Efficient implementation of quantum algorithms on hardware with limited qubit connectivity is a critical challenge in quantum computing. One-dimensional nearest-neighbor architectures restrict two-qubit interactions, making gate optimization essential for reliable computation.
In an n-qubit quantum circuit arranged in a one-dimensional nearest-neighbor configuration, which optimization most significantly reduces the error rates and improves efficiency when implementing the Quantum Fourier Transform (QFT)?
1) Increasing the number of single-qubit gates relative to two-qubit gates
2) Using teleportation protocols to bypass connectivity constraints
3) Replacing all CNOT gates with SWAP gates
4) Adding ancillary qubits to increase connectivity
5) Reducing the number of CNOT gates by approximately 60% through circuit redesign
6) Encoding qubits using higher-dimensional qudits
7) Implementing classical post-processing after each quantum gate operation
✓ Correct Answer:
The correct answer is 5) Reducing the number of CNOT gates by approximately 60% through circuit redesign.
📚 Reference Text:
Title: Optimizing the number of CNOT gates in one-dimensional nearest-neighbor quantum Fourier transform circuit Year: 2022 Paper ID: 45a58beed8bb61afbe7747ef75223b4289e6297b Source: semantic-scholar URL: https://www.semanticscholar.org/paper/45a58beed8bb61afbe7747ef75223b4289e6297b Abstract: : The physical limitations of quantum hardware often require nearest-neighbor qubit structures, in which two-qubit gates are required to construct nearest-neighbor quantum circuits. However, two-qubit gates are considered a major cost of quantum circuits because of their high error rate as compared with single-qubit gates. The controlled-not (CNOT) gate is the typical choice of a two-qubit gate for universal quantum circuit implementation together with the set of single-qubit gates. In this study, we construct a one-dimensional nearest-neighbor circuit of quantum Fourier transform (QFT), which is one of the most frequently used quantum algorithms. Compared with previous studies on n-qubit one-dimensional nearest-neighbor QFT circuits, it is found that our method reduces the number of CNOT gates by ~60% . Additionally, we showed that our results for the one-dimensional nearest-neighbor circuit can be applied to quantum amplitude estimation.
Question 2multiple-choice
Quantum error mitigation strategies are essential for improving the reliability of computations on noisy intermediate-scale quantum (NISQ) devices. The design and interaction requirements of such protocols strongly influence their suitability for current hardware platforms.
Which feature makes a quantum error mitigation protocol especially feasible for implementation on existing NISQ devices with limited qubit connectivity?
1) Requiring only nearest-neighbor interactions between qubits
2) Utilizing long-range, all-to-all qubit connectivity
3) Demanding specialized cryogenic hardware for error correction
4) Relying exclusively on superconducting qubits
5) Necessitating exponential overhead in initial state preparation
6) Depending on error correction tailored solely for phase-flip errors
7) Employing global entangling gates across the entire qubit array
✓ Correct Answer:
The correct answer is 1) Requiring only nearest-neighbor interactions between qubits.
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parity encoding is the belief propagation proposed in Ref. [ 49]. The effects of this fault tolerance on circuit execution results wasnot covered in this work. Our proposal requires interactions between nearest neigh- bors only and can thus be implemented on various currentNISQ devices with their natural interqubit connectivity. Theproposal is also independent of the specific qubit plat-form. Suitable platforms are, for example, superconducting 042442-6 APPLICATIONS OF UNIVERSAL PARITY QUANTUM … PHYSICAL REVIEW A 106, 042442 (2022) qubits [ 50–55], neutral atoms [ 13,14,56], or trapped ions [18,19,57,58]. In order to complement the bit-flip tolerance of the parity encoding, the use of noise-biased qubits [ 59–61] may be considered. A combination of our findings regarding the QFT in the LHZ scheme with the achievements presented in Refs. [ 62,63] on programming arbitrary superposition states using quantumannealers may give rise to a QFT device without exponentialgate overhead for the initial state preparation [ 31] and opens up a promising avenue for further research.ACKNOWLEDGMENTS We thank C. Ertler for valuable input on the graph state preparation and for numerous fruitful discussions. Work at the University of Innsbruck was supported by the Austrian Science Fund (FWF) through a START grant under ProjectNo. Y1067-N27 and the Special Research Programme (SFB)“BeyondC: Quantum Information Systems Beyond ClassicalCapabilities”, Project No. F7108-N38. This work was sup-ported by the Austrian Research Promotion Agency underGrant (FFG Project No. 892576, Basisprogramm). [1] D. Deutsch and R. Jozsa, Rapid solution of problems by quan- tum computation, Proc. R. Soc. London 439, 553 (1992) . [2] P. W. Shor, Polynomial-time algorithms for prime factoriza- tion and discrete logarithms on a quantum computer, SIAM J. Comput. 26, 1484 (1997) . [3] E. Bernstein and U. Vazirani, Quantum complexity theory, SIAM J. Comput. 26, 1411 (1997) . [4] L. K. Grover, Proceedings of the Twenty-Eighth Annual ACM Symposium on Theory of Computing, Philadelphia, 1996 (As- sociation for Computing Machinery, New York, 1996), pp.212–219 [5] M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, Experi- mental Realization of any Discrete Unitary Operator, Phys. Rev. Lett. 73, 58 (1994) . [6] A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVincenzo, N. Margolus, P. Shor, T. Sleator, J. A. Smolin, and H. Weinfurter,Elementary gates for quantum computation, P h y s .R e v .A 52, 3457 (1995) . [7] M. Mosca, Quantum computer algorithms, Ph.D. thesis, Uni- versity of Oxford, 1999. [8] R. Raussendorf and H. J. Briegel, A One-Way Quantum Com- puter, Phys. Rev. Lett. 86, 5188 (2001) . [9] R. Raussendorf, D. E. Browne, and H. J. Briegel, Measurement- based quantum computation on cluster states, P h y s .R e v .A 68, 022312 (2003) . [10] T. Albash and D. A. Lidar, Adiabatic quantum computation, Rev. Mod. Phys. 90, 015002 (2018) . [11] M. Fellner, A. Messinger, K. Ender, and W. Lechner, Univer- sal Parity Quantum Computing, Phys. Rev. Lett. 129, 180503 (2022) . [12] W. Lechner, P. Hauke, and P. Zoller, A quantum annealing architecture with all-to-all connectivity from local
Question 3multiple-choice
Quantum computing leverages superposition, interference, and entanglement to perform calculations fundamentally differently from classical computers. Measurement plays a unique role in quantum algorithms, impacting the reversibility and information content of quantum states.
Which of the following accurately describes the effect of measurement on a quantum system of multiple qubits?
1) Measurement applies a unitary transformation that preserves superposition and probability amplitudes
2) Measurement probabilistically amplifies all possible outcomes equally
3) Measurement converts quantum entanglement into classical correlations without loss of information
4) Measurement irreversibly collapses the superposed quantum state to a single basis state, with outcome probabilities determined by the squared magnitudes of amplitudes
5) Measurement reversibly projects the quantum state onto multiple basis states simultaneously
6) Measurement preserves the full quantum information and coherence of the system
7) Measurement deterministically transforms the quantum state into a preselected outcome
✓ Correct Answer:
The correct answer is 4) Measurement irreversibly collapses the superposed quantum state to a single basis state, with outcome probabilities determined by the squared magnitudes of amplitudes.
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combined to interfere producing the computation output. Qubits are manipulated by quantum gates, most of which represent unitary transformations. The general state of a single qubit is a unit vector in a two-dimensional Hilbert space and with a standard choice of the computational basis fj0i;j1igcan be written as jyi=aj0i+bj1i=0 B@a b1 CA; (1.1) where amplitudes aandbare complex numbers such that jaj2+jbj2=1 (1.2) The statejyiis in a superposition of the states j0iandj1i, i.e., it exists in both states simultane- ously. 1 The quantum state for a system of nqubits is a unit vector in a 2n-dimensional Hilbert space and can be expressed in the computational basis as jy(n)i=1 å i0=0:::1 å in 1=0ai0;:::;in 1ji0:::in 1i (1.3) wherejy(n)iis normalized. A system of nqubits represents a superposition of 2nstates, allowing a quantum computer to operate on states simultaneously. This phenomenon of quantum parallelism is the source of the quantum computer’s overwhelming power that allows for the execution of algorithms which no classical computer can efficiently handle. However, the only way to extract information from qubits is to subject them to measurement, which reduces the system to a single state in accordance with the Born rule. The rule states that for a qubit in a superposition (1.1) the measurement result is 0 with probability jaj2or 1 with probability jbj2. Thus, the post-measurement state no longer contains any information pertaining to the amplitudes of the pre-measurement superposition state, other than indicating that the particular amplitude corresponding to the actual output state was not 0, and likely not exceedingly small. Moreover, a measurement gate is not unitary, i.e., it is irreversible. For an input xcovering the range 0 x<2nand a computed function f, the measurement of the input register produces a single randomly chosen x0, while the measurement of the output register gives a single corresponding value f(x0). To take advantage of superposition and thus achieve an exponential speedup over the clas- sical computer, quantum algorithms involve judicial arrangements of unitary gates, in some cases supplemented by intermediate measurement gates acting on subsets of a system’s qubits. In most cases, this approach leads to an extraction of information about global properties of f, which man- ifest themselves in the relations between the values of ffor many distinct values of x, information that a classical computer can only produce by making many independent evaluations. 2 1.2 A Brief History of Quantum Computing The Turing machine is the standard mathematical model of a classical computer. A math- ematical model of a quantum computer, just as independent of a physical realization, was first discussed by Manin [1] in 1980. A few years later, Feynman [2, 3] pointed out that simulation of certain quantum systems on a classical computer requires time that scales exponentially with the number of particles in a simulated system. Feynman proposed that a quantum system might be efficiently simulated by a computer that exploits quantum effects. In 1985, a concrete mathematical model, the quantum Turing machine, was introduced by Deutsch [4], who followed it up in 1989 [5] by
Question 4multiple-choice
In quantum computing, the implementation of quantum gates often involves evolving a closed quantum system under a time-dependent Hamiltonian that includes both field-free and control components. The properties of the Hamiltonian and the gate generator play a crucial role in determining gate efficiency and phase characteristics.
Which statement correctly describes the effect of shifting the eigenvalues of a quantum gate's generator by integer multiples of 2π?
1) It changes the physical gate operation and the eigenbasis of the system.
2) It modifies the system's evolution operator and prevents unitary implementation.
3) It affects the local phases of individual computational basis states only.
4) It leaves the gate unchanged but alters the generator and its trace, impacting the global phase.
5) It always results in a non-traceless effective Hamiltonian incompatible with SU requirements.
6) It makes the gate implementation faster by reducing the critical time to zero.
7) It removes the need for control Hamiltonians in the implementation process.
✓ Correct Answer:
The correct answer is 4) It leaves the gate unchanged but alters the generator and its trace, impacting the global phase..
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GLOBAL PHASE AND EFFECTIVE HAMI LTONIAN A. Correlation of the global phase and the effective Hamiltonian of a gate We consider the problem of the implementation of a quantum gate in a closed quantum system with Hamiltonian 0 ( ) ( )ff fH t H u t H , (3) where 0H is the field-free Hamiltonian , fH is the f-th control Hamiltonian operator , and ()fut is the amplitude of the corresponding control field s. We have to find the control field s ()fut at which the operator of the system evolution for time T is 0ˆ ( ) exp ( )T U T T i H t dt (4) that performs the desired logic transformation specified by the unitary matrix U( )GUN in a certain computational basis . Here, ˆT is the time -ordering operator. Unitary gate GU can be presented in the exponential form exp( )GU iK . (5) For convenience, we take the negative exponent, by analogy with the definition of evolution operator (4). Using transformation P, we reduce matrices GU and K to the diagonal form † 1N k kP KP D k k , † 1exp( ) exp( )N Gk kP U P iD i k k , (6) where kk is the projector onto eigenstate k. Now, if we add the numbers 2km , where km is an integer, to one or several eigenvalues k , then the value of the exponential operator in Eq. (6) does not change, but matrix D change s and, consequently, matrix K is transformed to the new matrix, †† 1( ,..., ) 2 2m N k k kkK K m m P D m k k P K m P k k P (7) with the transformed trace 2m m k kTrK TrK m . (8) 4 To implement gate GU on the quantum system with the traceless Hamiltonian [i.e., ( ) SU( )U T N ], one should take the operator eff m m mTH K E , (9) as an effective Hamiltonian. Here, /mm N and Е is the identity operator . Substituting this expression in evolution operator definitio n (4), we obtain ( ) exp( ) exp( )eff m m m GU T iTH i U . (10) Comparing (10) and (1) , we obtain mod(2 )pm . Thus , different effective Hamiltonians (9) can lead to different global phase s (2). Moreover, it is reasonable to suggest that there exists a set of solutions of control task (4). Different solutions correspond ing to different eff mH can have different critical times. Therefore , one may choose the one from the set of effective Hamiltonians eff mH that has the required advantages, e. g., allows implement ing gate GU in a shorter period of time. Transformations (7) and (9) have a simple physical meaning . When in expression (7)
Question 5multiple-choice
In quantum many-body physics, calculating transport coefficients such as shear viscosity in strongly coupled gauge theories often requires sophisticated nonperturbative techniques. Recent advances in computational methods have enabled studies of these properties in low-dimensional lattice models.
In a (2+1)-dimensional SU(2) gauge theory studied on a small honeycomb lattice, which result for the shear viscosity to entropy density ratio (η/s) is consistent with predictions from gauge/gravity duality for strongly coupled quantum systems?
1) η/s = 0
2) η/s = 1
3) η/s = π
4) η/s = 1/4π
5) η/s = 2π
6) η/s = 1/2
7) η/s = 4/π
✓ Correct Answer:
The correct answer is 4) η/s = 1/4π.
📚 Reference Text:
Title: Classical and quantum computing of shear viscosity for (2+1)D
Question 6multiple-choice
In quantum error correction, duality transformations and encoding schemes are often used to construct codes with desirable properties, such as locality preservation and covariance under certain Hamiltonians. The Ising spin chain serves as a key example where these techniques enable robust and analyzable codes relevant to quantum computing and physics.
Which property is achieved by mapping the Ising spin chain Hamiltonian to a noninteracting Hamiltonian via a duality transformation and encoding, as described for quantum error-correcting code construction?
1) The code becomes immune to all possible single-qubit errors.
2) The code distance is doubled relative to the original model.
3) The Hamiltonian terms must commute for error correction to be analyzed.
4) Local operators in the original model map to similarly local operators in the transformed space.
5) The code can universally correct any number of erasures regardless of their location.
6) The worst-case error scaling becomes exponential in system size.
7) The encoding prevents any logical operator from acting nontrivially on the code space.
✓ Correct Answer:
The correct answer is 4) Local operators in the original model map to similarly local operators in the transformed space..
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properties of the code. To construct the code formally, we observe that, under a well-known duality transformation [61,62] , the Ising spin chain Hamiltonian (51) is equivalent to the noninteracting Hamiltonian ˜H¼Xn i¼2σZ i; ð52Þ which, except for the first site i¼1, is the same as the magnetization operator considered in our discussion of thethermodynamic code in Sec. VI C. This correspondence is established directly on the basis states fj ⃗xig ¼ fj x 1i⊗ /C1/C1/C1⊗jxnigof the full Hilbert space, where ⃗xis a bit string and the states jxi¼0;1iare eigenstates of the σZ ioperator. We encode this bit string into another bit string ⃗sð ⃗xÞthat provides the value of the first bit and all pairwise consecutiveparities, namely, s 1¼x1andsi¼xi−1þxiðmod 2Þfor 2≤i≤n. Clearly, this mapping is one to one and onto acting on the set of all bit strings of length n.D e n o t eb y U the unitary on the n-site Hilbert space that implements this transformation, i.e., U¼P ⃗xj ⃗sð ⃗xÞih ⃗xj.W es e et h a t σZ ij ⃗sð ⃗xÞi ¼ ð−1Þsij ⃗sð ⃗xÞ i¼ð−1Þxi−1þxij ⃗sð ⃗xÞi and, thus, U†σZ iU¼σZ i−1σZ i. This result implies that U†˜HU ¼HIsing.CONTINUOUS SYMMETRIES AND APPROXIMATE QUANTUM … PHYS. REV . X 10,041018 (2020) 041018-15 The Hamiltonian ˜His exactly the same operator as the magnetization operator in the thermodynamic code ofSec. VI C on the n−1the sites labeled by i¼2;…;n, with a dummy site at i¼1. Using the code words j0i 1⊗ jhn−1mi2…nwhich force the dummy site to the constant state j0i, we still have a code with the same asymptotic proper- ties as before (the dummy site cannot impact negatively theerror-correction properties of the code). Similarly, anyoperator supported on d−1consecutive sites is mapped under the unitary Uto an operator that has support on at most dsites. Hence, the erasure of d−1consecutive sites in the mapped model corresponds to erasure of at most d consecutive sites in the original model, which is correct-able. Our construction, thus, defines a code that is covariantwith respect to the Ising Hamiltonian, with distance d−1 and with ϵ worstðN∘EÞ≤Oð1=nÞ. Applying our bound directly to the newly constructed code, where any d−1consecutive sites can be erased with probability 1=ðn−dþ2Þ≈1=n, yields as before ϵworstðN∘EÞ≥Ωð1=nÞ. Although the overlapping terms in Eq. (51)all commute pairwise, we emphasize that Theorem 2 does not requirethese terms to commute. In consequence, one can study theerror-correction accuracy of codes that are defined usingmore complicated many-body models [8], going beyond the rather simplistic model presented here. Being able to characterize the performance of codes in which subsystems interact is helpful for potential applica-tions of our results. For example, in quantum metrology, theprobe detecting a weak signal might contain many mutuallyinteracting particles. In the AdS/CFT correspondence (seeSec. IX), the charge is the energy of a conformal field theory; in a lattice regularization of this theory, thesubsystems are lattice sites with strong nearest-neighbor interactions. VII. CODES WITH A UNIVERSAL TRANSVERSAL GATE SET Our second main technical contribution is a robust version of the Eastin-Knill theorem
Question 7multiple-choice
In quantum computing, the hidden subgroup problem (HSP) is a central challenge whose efficient solutions can unlock powerful algorithms for various group-theoretic problems. Techniques such as Gröbner bases, entangled measurements, and pretty good measurement (PGM) have played important roles in advancing quantum algorithms for HSP, especially over nonabelian groups.
Which group structure allows the hidden subgroup problem to be solved efficiently in poly(log p) time on a quantum computer, given r is fixed, and is directly connected to efficient Gröbner basis computation and quantum sampling?
1) Dihedral groups D_n
2) Cyclic groups Z_p
3) Abelian groups of order p^n
4) Zn_p ⋊ Z2 groups
5) Symmetric groups S_n
6) Zr_p ⋊ Zp groups with fixed r
7) General non-semidirect product groups
✓ Correct Answer:
The correct answer is 6) Zr_p ⋊ Zp groups with fixed r.
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our purposes, a Gr¨ obne r basis can be computed efficiently. Since we consider only the cases in which there are at most 2 so lutions, elimination is also efficient, giving an overall poly(log p) time algorithm for computing a list of matrix sum solutions , and hence for quantum sampling from those solutions. Collecting these results, we find Theorem 6. The hidden subgroup problem over any group of the form Zr p⋊Zpwithrfixed can be solved in time poly(logp)on a quantum computer. 7 Discussion In this paper, we have studied the pretty good measurement fo r semidirect product groups of the formA⋊ZpwithAabelian and pprime. We found that the pgmis closely connected to the matrix sum problem, and we exploited this connection to find efficient quantum algorithms for certain metacyclic groups (Section 5) as well as all groups of the for mZr p⋊Zpwithrfixed (Section 6). The latter algorithm demonstrates that entangled measurement s may be useful for efficiently solving the nonabelian hsp. Aside from the fact that these particular nonabelian groups admit efficient quantum algorithms, our results suggest two general directions for further inve stigation. First, in the standard approach to the nonabelian hsp, it would be helpful to have a better understanding of when en tangled measurementsarenecessaryandwhentheycanbeimplemented togiveefficientalgorithms. Second, for thehspor for other problems that can be viewed as quantum state dist inguishability problems, identifying an optimal measurement (or considering a parti cularly nice measurement such as the pgm, which in general may or may not be optimal) can be used as a pri nciple for discovering new quantum algorithms. While the reduction of Lemma 1 combined with the pgmapproach outlined in Section 4 appears to efficiently solve the hspin most of the semidirect product groups where efficient algor ithms are known, thereisonenotableexception. Thegroups Zn p⋊Z2(forwhichanefficient quantumalgorithm is given in[10]) give risetoasubsetsumproblemover Zn p, which appearstobeessentially as difficult as the subset sum problem over ZNarising from the dihedral group. Thus, it would be interesti ng to understand what allows the algorithm in [10] to be efficient even though the matrix sum problem is (apparently) hard. Of course, there are many nonabelian groups that are semidir ect products of nonabelian groups, orthatcannotbenontrivially decomposedintosemidirectp roductsatall. The pgmapproachiswell 14 defined for any group, so it would be interesting to explore th e approach in such cases, regardless of whether the pgmis optimal. Acknowledgments We thank Carlos Mochon and Frank Verstrate for helpful discu ssions of Theorem 3. AMC received support from the National Science Fo undation under Grant No. EIA- 0086038. References [1] N. Alon and J. H. Spencer. The Probabilistic Method . Wiley Interscience, New York, 2nd edition, 2000. [2] D. Bacon, A. M. Childs, and W. van Dam. Optimal measuremen ts for the dihedral hidden subgroup problem. arXiv:quant-ph/0501044. [3] R. Beals. Quantum computation of Fourier transforms ove r symmetric groups. In Proc. 29th Annual ACM Symposium on Theory of Computing , pages 48–53, 1997. [4] D. Boneh and R. J. Lipton. Quantum cryptanalysis of hidde n linear functions. In Lecture Notes
Question 8multiple-choice
In quantum information theory, the classification of multipartite entangled states relies on both representation theory and sophisticated geometric frameworks. Understanding how group actions partition state spaces is central to this classification.
Which statement accurately describes the role of the local unitary group SU(d₁) ×.. × SU(dₙ) when acting on the complex projective space CP^{D-1} of quantum states for a system with n subsystems of dimensions (d₁,.., dₙ)?
1) It identifies states that differ only by global phase, resulting in a set of orthogonal basis states.
2) It produces a decomposition of the Hilbert space into irreducible subspaces corresponding to each subsystem.
3) It combines all possible tensor product states into a single orbit representing total entanglement.
4) It partitions the projective space into orbits, each corresponding to a distinct entanglement class invariant under local unitary transformations.
5) It restricts physical states to those containing the identity representation in their tensor products.
6) It transforms entangled states into separable states through normalization and phase adjustment.
7) It defines the dimension of the projective space by the product of subsystem dimensions.
✓ Correct Answer:
The correct answer is 4) It partitions the projective space into orbits, each corresponding to a distinct entanglement class invariant under local unitary transformations..
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database. Forn≥4 it is straightforward to see that sufficiency can’t hold. To see this, note that if there exist representations with dimensions {d1,d2,d3,d4}whose tensor product contains the identity, then the tensor product of any two of the represen tations must contain a rep- resentation whose dimension is the same as a representation in the t ensor product of the other two representations. But this cannot work for dimensions ( 2,2,2,7) (for which LME states exist), since 2 ×2 can contain no representation of dimension larger than 4, while fro m Corollary 3.6, we find that 2 ×7 can contain no representation of dimension smaller than 6. 4 Background: The geometry of LME states In this review section, we describe how the set of LME states has tw o natural geometrical formulations that turn out to be equivalent to each other. The firs t is related to symplec- tic geometry and the other is related to algebraic geometry and geo metric invariant theory. Physically, these two perspectives relate to two seemingly different classification problems for quantum states. This material has been discussed previously in the literature; see for example [Br02], [BB02], [Kly02], [Kly07, §3], [GW10], [SOK12], [Wall08, §4], [Walt14] and [SOK14]. Additional reviews on the classification of multipart entangle ment and the geom- etry of multipart Hilbert spaces can be found in [ES14] and [BZ17], res pectively. Readers already familiar with these geometrical preliminaries can skip directly t o section 5. 22 4.1 Geometry of the space of states We begin with the full space of (unnormalized) states. For dimension s (d1,...,dn), the Hilbert space H=H1⊗···⊗H nis the complex vector space CDwithD=d1d2···dnwhose coordinates can be taken as the coefficients ψi1...indefining the state. The inequivalent physical states can be represented as equivalence classes of vec tors with unit norm with two states identified if they are multiplicatively related by a phase. Eq uivalently, we can work with unnormalized states, omitting the zero vector and identif ying states related via multiplication by a complex scalar. This defines the complex projective spaceP(H) = CPD−1. 4.1.1 Entanglement structure and the action of K= SU(d1)×···×SU(dn) It is an interesting question to characterize the possible entanglem ent structures that such states can have. By “entanglement structure” we mean propert ies of a state that are un- affected by local unitary transformations; that is, unitary trans formations that act inde- pendently on each tensor factor of the Hilbert space. These tran sformations correspond to changes of basis for the individual subsystems. Mathematically, th e change of basis oper- ations correspond to the group ˜K=U(d1)× ··· ×U(dn) acting on H. Without loss of generality, we may consider the smaller group K= SU(d1)×···×SU(dn) since the action of˜KandKon the projective space P(H) =CPD−1have the same orbits. Each state in P(H) will be on some orbit of K. The space of these orbits then represents the space of possible entanglement structures. To parameterize the space of these orbits, we can define a set of coordinates which are polynomials in ψandψ†invariant under the
Question 9multiple-choice
Quantum algorithms often rely on amplitude amplification and the Quantum Fourier Transform (QFT) to solve problems in cryptography and computational complexity. Ensuring that these algorithms succeed with certainty, rather than high probability, is a major technical challenge.
Which of the following techniques enables amplitude amplification to work exactly (with guaranteed success), even when the success probability of eigenvalue estimation depends on the input instance?
1) Increasing the number of qubits in the quantum register
2) Performing the Quantum Fourier Transform twice in succession
3) Uniformizing the success probability by randomly shifting the input state and correcting afterwards
4) Measuring the quantum state after every operation
5) Using classical post-processing to identify correct eigenvalues
6) Applying an unstructured Grover search to the problem
7) Replacing phase estimation with brute-force search algorithms
✓ Correct Answer:
The correct answer is 3) Uniformizing the success probability by randomly shifting the input state and correcting afterwards.
📚 Reference Text:
eiϕto the good subspace and leave its orthogonal complement unchan ged. So how can we check whether a number x′is the right eigenvalue of |Ψx/an}⌊ra⌋k⌉tri}ht, thus whether x′=x? This can be done because the eigenstate |Ψx/an}⌊ra⌋k⌉tri}htis still available exactly. Thus given a state of the form |Ψx/an}⌊ra⌋k⌉tri}ht/summationtext x′cx,x′|x′,gx,x′/an}⌊ra⌋k⌉tri}ht, we can check the second register against the first one. To do this we apply the re verse of the steps in eq. 1 to these two registers, thus: |x′,Ψx/an}⌊ra⌋k⌉tri}ht → | x′,Ψx−x′/an}⌊ra⌋k⌉tri}ht → | x′,θx−x′/an}⌊ra⌋k⌉tri}ht where in the second step we only act on the second register. The st ate|Ψ0/an}⌊ra⌋k⌉tri}htis mapped back to |0/an}⌊ra⌋k⌉tri}ht, while for x′/n⌉}ationslash=xwe get some state |θx−x′/an}⌊ra⌋k⌉tri}htorthogonal to |0/an}⌊ra⌋k⌉tri}ht. We can now apply the phase eiϕto the|0/an}⌊ra⌋k⌉tri}htstate and undo the previous operations. 3 2.2 “Uniformising” the success probability One obstacle to using amplitude amplification to make algorithms exact is that the success probability of the “heuristic” algorithm Amust be known. But this probability may depend on the (unknown) instance of the problem. I n our case the success probability of eigenvalue estimation on |Ψx/an}⌊ra⌋k⌉tri}htindeed does depend on x. Wecanfix this problembymodifying Asuchthatthe new successprobability will become instance independent and equal to the average over all instances for the original A. To do this uniformisation we pick an integer runiformly at random from {0,1,...p−1}and replace |Ψx/an}⌊ra⌋k⌉tri}htwith|Ψx+r/an}⌊ra⌋k⌉tri}ht, which is just a rephasing. We keep a record of rand subtract it again from the result of eigenvalue estimation. To do this with a unitary Awe will need an additional register for r, but this is no problem, as we have already included the possibility that eigenvalue estimation (eq. 2) also generates some unwanted g arbagegx,x′. So now exact amplitude amplification will allow us to do |Ψx,0/an}⌊ra⌋k⌉tri}ht → | Ψx/an}⌊ra⌋k⌉tri}ht|x,gx,x/an}⌊ra⌋k⌉tri}ht. To get rid of the “garbage” we can do the usual trick of copying the wanted resultxinto an additional “save” register and then undoing the previous st eps. In total this will lead to six applications of Afor an exact QFFT. In summary, the construction of an exact QFFT relies on making eige nvalue estimation (on Fourier states |Ψx/an}⌊ra⌋k⌉tri}ht) exact. The essential observations are that eigenvalue estimation leaves the eigenstate |Ψx/an}⌊ra⌋k⌉tri}htexactly unchanged and so it can be used for the checking stage of amplitude amplification. Furth ermore we used that the success probability of estimating xfrom|Ψx/an}⌊ra⌋k⌉tri}htcan rather easily be “uniformised” across all x= 0...p−1. 3 An exact discrete logarithm algorithm An exact algorithm for the QFFT leads in a straightforwardmanner t o an exact algorithm for the discrete logarithm algorithm of the same order. Th is was also observed for finite fields of prime order by Brassard and Høyer [2] (Theorem 12). For smooth orders (only small prime factors) the problem can easily be solved classically. Here we give a quick review for the case when the or der is a large prime (see also [12], section 2.2.3). In a discrete logarithm problem we are given an element αwhich generates a cyclic group of some finite order, here a prime. Thus
Question 10multiple-choice
In liquid-state NMR quantum computing, precise modeling of multi-spin molecular systems is essential for accurate simulation of quantum algorithms and interpretation of experimental results. Understanding how different nuclear spins interact and relax within a molecule is a key aspect of this domain.
When simulating the quantum dynamics of an alanine molecule for NMR quantum computing experiments, which approach correctly accounts for the influence of hydrogen spins on the carbon-13 qubits during a Quantum Fourier Transform experiment?
1) Hydrogen spins are actively flipped during the experiment and require time-dependent modeling in the Hamiltonian.
2) Hydrogen spins are omitted entirely from the simulation because their couplings are negligible.
3) Hydrogen spins are included as coherent superpositions in the initial density matrix for the carbon system.
4) Hydrogen spins influence carbon-13 qubits only through dipolar couplings that must be averaged over molecular motion.
5) Hydrogen spins are treated as constants of motion, resulting in an incoherent mixture of 16 independent 3-spin Hamiltonians for the carbons, each corresponding to a different hydrogen configuration.
6) Hydrogen spins are modeled by their effect on the nitrogen-14 quadrupolar relaxation only.
7) Hydrogen spins are assumed to have the same relaxation dynamics as carbon-13 qubits and contribute equally to decoherence.
✓ Correct Answer:
The correct answer is 5) Hydrogen spins are treated as constants of motion, resulting in an incoherent mixture of 16 independent 3-spin Hamiltonians for the carbons, each corresponding to a different hydrogen configuration..
📚 Reference Text:
Hamiltonian. •The Hilbert space of the larger system with which the given system int eracts coher- ently, and the Hamiltonian describing these interactions. •The relaxation superoperator of the system, and some knowledge of how the larger system relaxes. •Bounds on the precision of the classical control fields applied. 12 •The distribution of incoherent variations in these fields across the s ample. Both the fields and the positions of the spins can be treated classica lly. Since the larger system contains only a total of eight spins, we can simulate it exactly . Further information, for example the correlation times or full spectral densities of the n oise generators driving relaxation [32], could have been included in the model, but the above pr oved adequate to explain most of the experimental observations for the QFT. The first attribute is the Hamiltonian for the three carbon-13 spins of alanine used as qubits for theQFT, which hasthe formof Hsgiven in Eqn. (19). The larger system includes, in addition to the carbons, the four (nonexchanging) hydrogens a nd the spin-1 nitrogen-14 nucleus in alanine. The larger system’s Hamiltonian has the form HS=Hs+π4/summationdisplay i=1νi Hσi H,z+πνNσN,z+π 24/summationdisplay j>i=1Ji,j H,H/vector σi H·/vector σj H+ (22) π 24/summationdisplay i=13/summationdisplay j=1Ji,j C,Hσi H,zσj C,z+π 23/summationdisplay i=1Ji N,CσN,zσi C,z+π 24/summationdisplay j=1Jj N,HσN,zσj H,z, where we have used the well-known fact that couplings between spin s with distinct gyro- magnetic ratios can be truncated to just the secular (i.e. σzσz) part. The spin-1 nitrogen has an electric quadrupole moment and hence a s hortT1, so its coupling to the other spins cannot be observed. As a result, the nit rogen can be omitted from the Hamiltonian, although it still plays a role in the relaxation of th e other spins. The spin-lattice ( T1) relaxation time of the hydrogen atoms, on the other hand, is longe r than the experiment, so here we must include the additional frequency s hifts that depend upon their spin states. In other words, we take the σH,zto be constants of the motion, and treat the carbons as an incoherent mixture of 24= 16 independent 3-spin systems, each with their resonance frequencies shifted by one of the 16 possible sums of±JC,H/2. Because we start with the spins in the high-temperature equilibrium state, each of these 16 independent evolutions contributes equally to the simulated density matrix. The computer search for the strongly modulating control sequen ces (pulses) is quite demanding, and it is important to keep the Hilbert space as small as po ssible during the associated simulations. For this reason the simulations described in t his paper ignored the four hydrogen atomsinthe alaninemolecule, which were left alignedwit h themain magnetic field during the actual experiments but which nonetheless have cou plings to the carbons on 13 the order of 150Hz. A better way, which we subsequently implement ed and is described here for completeness, would be to average the results of 16 simulations over each of the 3-spin Hamiltonians Hδ s,c=π3/summationdisplay i=1/parenleftBig νi C+4/summationdisplay j=1(−1)δj
Question 11multiple-choice
In computational molecular spectroscopy, various quantum chemical and simulation techniques are used to predict vibrational frequencies and interpret solvent effects on small molecule spectra. The interplay between theoretical methods and experimental measurements is critical for understanding hydrogen bonding and molecular interactions in aqueous environments.
Which computational approach utilizes a composite of CCSD reference frequencies with a QM(AM1)/MM perturbation potential to achieve both high accuracy and computational efficiency in predicting vibrational spectra?
1) QVP1 method
2) Dipole Autocorrelation Function Fourier transform
3) Harmonic Normal Mode Analysis with HF basis
4) Spectral Expansion using MP2 reference
5) DAF using CCSD exclusively
6) QM/MM simulation with AM1 only
7) Two-dimensional vibrational Schrödinger equation on M06-2X surface
✓ Correct Answer:
The correct answer is 1) QVP1 method.
📚 Reference Text:
(black) and computed vibrational spectra of the carbonyl stretching region of acetone in water. The spectrum from the first-order quantum vibration perturbation (QVP1) calculation employing a composite of the CCSD(T) reference frequency and a QM(AM1)/MM perturbation potential is shown in red, and that from the Fourier transform of the molecular dipole autocorrelation function determined using AM1 directly in QM/MM simulations is given in blue.Xue et al. Page 24 J Chem Theory Comput . Author manuscript; available in PMC 2018 February 01. Author Manuscript Author Manuscript Author Manuscript Author Manuscript Author Manuscript Author Manuscript Author Manuscript Author ManuscriptXue et al. Page 25 Table 1 Computed HCl Stretching Vibrational Frequency ωCl−H and the Shift ΔωCl−H of the HCl(H 2O) Complex Using Harmonic Normal Mode Analysis (NMA), Fourier Transform of the Dipole Autocorrelation Function (DAF), and the Spectral Expansion (eq 6) of the Frequency-Frequency Time Correlation Function from the QVP1 Method, along with the Experimental Value (Vibrational Frequencies Given in cm−1) method basis set ωCl−H ΔωCl−Ha HFQb 3033.0 309.5 NMA15 MP2Tb 2791.0 67.5 CCSD(T)Db 2782.0 58.5 DAF M06-2X cc-pVTZ 2837.6 114.1 QVP1 M06-2X cc-pVTZ 2653.0 −70.5 M06-2X/CCSD(T)-F12bcc-pVTZ/cc-pVTZ-F12c2724.8 1.3 CCSD(T)-F12b cc-pVTZ-F12 2737.6 14.1 EXP15,16 2723.5 aΔωCl−H = ωCl−H − ωexp. baug-cc-pVXZ, X = D,T,Q. cM06-2X/cc-pVTZ+CCSD(T)-F12b/cc-pVTZ-F12. J Chem Theory Comput . Author manuscript; available in PMC 2018 February 01. Author Manuscript Author Manuscript Author Manuscript Author ManuscriptXue et al. Page 26 Table 2 Computed H−Cl Stretching Vibrational Frequencies (cm−1) for (HCl) 2(H2O) by Solving Two Independent One-Dimensional (E 1D) Schrodinger Equations and a Two-Dimensional (E 2D) Vibrational Schrodinger Equationa Mode E1D E2D Δν Q1 2617.0 2612.3 4.7 Q2 2692.6 2693.7 −1.1 aThe potential energy surfaces were determined using M06-2X/cc-pVTZ. Q 1 and Q 2 represent, respectively, the H−Cl stretch associated with the HCl molecule directly hydrogen-bonded to water and hydrogen-bonded to the probe HCl molecule (Figure 5). J Chem Theory Comput . Author manuscript; available in PMC 2018 February 01.
Question 12multiple-choice
Photonic quantum processors utilizing qudits—quantum systems with more than two levels—are advancing the scalability and flexibility of quantum computing. These devices often integrate multiple quantum logic gates, entanglement sources, and precise optical components on a single chip.
Which feature uniquely enables a photonic multiqudit quantum processing unit to achieve fully reconfigurable and programmable initialization, manipulation, and analysis of multidimensional qudit states?
1) Integration of only polarization-encoded photon sources
2) Use of superconducting nanowire single-photon detectors
3) Implementation of time-bin encoding exclusively
4) Utilization of only single-qubit gates on a photonic chip
5) Limitation to probabilistic two-level entanglement
6) Absence of on-chip phase-shifters
7) Integration of arbitrary state preparation, multiqudit controlled logic gates (MVCU), and measurement for single and two-qudit states
✓ Correct Answer:
The correct answer is 7) Integration of arbitrary state preparation, multiqudit controlled logic gates (MVCU), and measurement for single and two-qudit states.
📚 Reference Text:
e 2shows the core of a multiqudit processor, i.e, the multiqudit multi-value con- trolled logic gate, which is realis ed by the following three steps: generation of the multiphoton multidimensional Greenberger- Horne-Zeilinger entangled state GHZ ji nþ1;d45,46,w h i c he n a b l e st h e entangling operations between the multiqudit states; Hilbert space expansion of each qudit in y-register to form an entire space of d2n, that locally allows individual and ar bitrary single-qudit operations56; coherent compression of the entire state back to a dnspace57.T h e s e sequences of operations result in a mu ltiqudit multi-value controlled- unitary (MVCU) gate as1ffiffi dp∑d/C01 j¼0kj/C12/C12/C12E /C10Qn i¼1Oi;jϕ/C12/C12/C11 i,w h e r e kj/C12/C12/C12E in the auxiliary x-register presents the logical state in the j-th mode (for simplicity it is denoted as j/C12/C12/C11 ), and Oi,jin the data y-register refers to an arbitrary local o p e r a t i o no nt h eq u d i ts t a t e ϕ/C12/C12/C11 ithat is initialised by the Piqudit generator. Such multiqudit MVCU gate works with a (1/ d) success probability regardless of n(see Supple- mentary Note 3 and Supplementary F ig. 1). The quantum circuits in Fig.2a, b provide a scheme of implementing multiqudit quantum Fourier algorithms in the scalable Kitaev ’s framework26–29. Figure 2c illustrates the integrated photonic quantum circuits for a two-ququart version of qudit-based quantum processing unit ( d- QPU). It was fabricated in silico n using the complementary metal- oxide-semiconductor (CMOS) process with the 248nm deep ultraviolet lithography (see a device image in Fig. 2d). The processor allows the generation of a path-enc oded two-ququart entangled state of GHZji2;4(i.e., the 4-dimensional generalised Bell state of Bell ji4), by a coherent excitation of four integrated spontaneous four-wave- mixing (SFWM) sources. It is followed by the sequences of processes of“space expansion –local operation –coherent compression" for the realisation of d-QPU, see Fig. 2b. The d-QPU chip monolithically integrates the core capabilities and fu nctionalities, including arbitrary single-ququart preparation ( P), arbitrary two-ququart MVCU operation (that presents a d-ary generalisation of two-qubit controlled-unitary operation), and arbitrary single-ququart measure- ment ( M). Though on-chip generatio n, manipulation and measure- ment of entangled qudit states have been reported36,t h i sw o r k demonstrate the key abilities to in itialize, manipulate, and analyze qudit states and gates in a fully recon figurable and reprogrammable manner, providing a major technological advance for qudit quantum computing. In Fig. 2d it shows one of the largest-scale programmable quantum photonic chip having 451 photonic components, including 116 recon figurable phase-shifters (see their characterisations in Fig. 2c i n s e t s ) .T h et w o - p h o t o n sd e t e c t i o nr a t ea tt h em a g n i t u d eo f1 03/s was measured in the two-ququart device, which is six orders higher than that in a four-qubit device (note
Question 13multiple-choice
In quantum computing, the Hidden Subgroup Problem (HSP) is central to the development of efficient algorithms and involves distinguishing between quantum states associated with different subgroups. Measurement strategies, such as the Pretty Good Measurement (PGM), play a crucial role in solving HSPs, especially for groups with special symmetries.
Which measurement strategy is proven to be optimal for identifying hidden subgroups when the subgroups are conjugate to a fixed subgroup and are sampled uniformly?
1) Maximum Likelihood Measurement
2) Projective Measurement onto Irreducible Representations
3) Pretty Good Measurement (PGM)
4) Von Neumann Measurement
5) Adaptive Bayesian Measurement
6) Swap Test Measurement
7) Holevo-Helstrom Measurement
✓ Correct Answer:
The correct answer is 3) Pretty Good Measurement (PGM).
📚 Reference Text:
generalized measurement can be described by a set of positive operators that sum to identity. If EHis the measurement operator corresponding to our guessing that the hidden subgroup is H, then the optimal measurement problem in the single copy case maxim izes the probability of successfully identifying the subgroups, psucc=/summationdisplay H∈Sub(G)pHTr[ρHEH], (4) subject to EHbeing a valid generalized measurement: EH≥0 and/summationtext H∈Sub(G)EH=I. Here Sub( G) is the set of subgroups of G. This is the well-known problem of state distinction. The state distinc tion problem was considered in the seventies by Holevo [20], and Yuen, Kennedy, and Lax [21] whe re necessary and sufficient conditions for the optimal measurement were derived for the generic state distinctio n problem. Applied to the above formulation of the hidden subgroup problem, these necessary and sufficient condition s yield that a measurement EHis optimal if and only if /summationdisplay H∈Sub(G)pHEHρH=/summationdisplay H∈Sub(G)pHρHEH, /summationdisplay H∈Sub(G)pHEHρH≥pH′ρH′,for allH′∈Sub(G). (5) We may also consider the multi-copy optimal measurement, where we query the function rtimes. In that case we produce rcopies of the hidden subgroup state ρH, and the optimality condition is obtained by simply replacing ρH byρ⊗r Hin Eq. (5). In [15], Ip considered the question of identifying the optimal measur ement for the abelian HSP when each of the subgroups of Gis given with equal a priori probability. Here we briefly review this resu lt. Define PH:=|G| |H|ρH. (6) The optimal single copy measurement for the abelian HSP is then given by the measurement operators (recursively defined) EH:=PH−/summationdisplay J⊃HEJ. (7) 3 In the case of rcopies, the optimal measurement is given by the similar expression E(r) H:=P⊗r H−/summationdisplay J⊃HE(r) J. (8) Finally Ip also showedthat for the HSP overthe dihedral group, the aboveexpressionis not the optimal measurement. Following Ip’s work, Bacon, Childs, and van Dam considered the optima l measurement for the HSP over the dihedral [16] and the Heisenberg groups [14]. In both of these case s, the measurement turned out to be the pretty good measurement (PGM), so named because it does a good job of id entifying states [17]. If the state ρiis given with probability pi, then the PGM is a measurement consisting of measurement operat orsEidefined as Ei:=piM−1/2ρiM−1/2, M:=/summationdisplay ipiρi, (9) where the inverse square root M−1/2is taken over the support of M. Bacon, Childs, and van Dam showed that the PGM was optimal for the dihedral and Heisenberg HSPs for both the single and multi-copy case and when the subgroups were restricted to certain order 2 or order psubgroups (which is enough to be able to solve the HSP over all subgroups due to a generalization of a theorem of Ettinger and H øyer [14, 16, 22].) Finally, Moore and Russell showed that under certain further cond itions the PGM is guaranteed to be optimal [18]. In particular they showed that if the hidden subgroup is restricted to come from the set of subgroups conjugate to a fixed subgroup, then for the single copy case, if these subgroups are sampled with uniform
Question 14multiple-choice
In Group Field Theory, quantum gravity condensates are studied as analogues of Bose-Einstein condensates, aiming to understand how macroscopic spacetime geometry can emerge from microscopic quantum building blocks. The behavior of condensate population and the mathematical representation of their quantum states are crucial in analyzing different interaction regimes.
What is a key indication that non-Fock representations are required to accurately describe quantum gravity condensates in Group Field Theory?
1) The presence of only combinatorially nonlocal interaction terms
2) The unbounded growth ("blow-up") of the condensate population number operator in the strong interaction regime
3) The dominance of the highest geometric configuration in operator spectra
4) The invariance of condensate population across all interaction strengths
5) The vanishing expectation values of geometric operators for all solutions
6) The absence of any interaction terms in the model
7) The exclusive validity of the Bogoliubov ansatz in strong nonlinear regimes
✓ Correct Answer:
The correct answer is 2) The unbounded growth ("blow-up") of the condensate population number operator in the strong interaction regime.
📚 Reference Text:
Title: Impact of nonlinear effective interactions on group field theory quantum gravity condensates Year: 2016 Paper ID: e891fbef0ff63acf937f4ffad2aba7cbb7a70311 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/e891fbef0ff63acf937f4ffad2aba7cbb7a70311 Abstract: We present the numerical analysis of effectively interacting Group Field Theory (GFT) models in the context of the GFT quantum gravity condensate analogue of the Gross-Pitaevskii equation for real Bose-Einstein condensates including combinatorially local interaction terms. Thus we go beyond the usually considered construction for free models. More precisely, considering such interactions in a weak regime, we find solutions for which the expectation value of the number operator N is finite, as in the free case. When tuning the interaction to the strongly nonlinear regime, however, we obtain solutions for which N grows and eventually blows up, which is reminiscent of what one observes for real Bose-Einstein condensates, where a strong interaction regime can only be realized at high density. This behaviour suggests the breakdown of the Bogoliubov ansatz for quantum gravity condensates and the need for non-Fock representations to describe the system when the condensate constituents are strongly correlated. Furthermore, we study the expectation values of certain geometric operators imported from Loop Quantum Gravity in the free and interacting cases. In particular, computing solutions around the nontrivial minima of the interaction potentials, one finds, already in the weakly interacting case, a nonvanishing condensate population for which the spectra are dominated by the lowest nontrivial configuration of the quantum geometry. This result indicates that the condensate may indeed consist of many smallest building blocks giving rise to an effectively continuous geometry, thus suggesting the interpretation of the condensate phase to correspond to a geometric phase.
Question 15multiple-choice
In the study of finite 2-groups, the structure of derived subgroups and maximal subgroups is crucial for understanding group classification, especially when certain subgroups are abelian or metacyclic. The relationship between the Frattini subgroup, derived subgroup, and properties of involutions often determines the possible isomorphism types of these groups.
Which of the following statements accurately describes the derived subgroup G' of a finite nonabelian 2-group G with three abelian maximal subgroups and abelian quotient types, given that G is not metacyclic?
1) G' is cyclic of order 8 and equals the center of G.
2) G' is elementary abelian of order 8 and disjoint from the Frattini subgroup Φ.
3) G' is trivial, implying G is abelian.
4) G' is contained in every maximal subgroup but not in the Frattini subgroup.
5) G' is isomorphic to C₄ and equal to the commutator subgroup of every maximal subgroup.
6) G' is isomorphic to C₂ × C₂ (elementary abelian of order 4) and is contained in the Frattini subgroup Φ.
7) G' is metacyclic and has order 2, coinciding with the center of G.
✓ Correct Answer:
The correct answer is 6) G' is isomorphic to C₂ × C₂ (elementary abelian of order 4) and is contained in the Frattini subgroup Φ..
📚 Reference Text:
≤Aso that H1//an}bracketle{tz/an}bracketri}htandH2//an}bracketle{tz/an}bracketri}htare two distinct FINITE 2-GROUPS 67 abelian maximal subgroups in G//an}bracketle{tz/an}bracketri}ht. It follows that H//an}bracketle{tz/an}bracketri}htis also abelian and soH′=/an}bracketle{tz/an}bracketri}htsinceHis nonabelian. If d( H) = 2, then (by Proposition 1.1) H would be minimal nonabelian, a contradiction. Thus d( H)≥3. IfG//an}bracketle{tz/an}bracketri}htis abelian, then G′=/an}bracketle{tz/an}bracketri}htand so G=H1CG(H1) which gives d(G) = 3, a contradiction. Hence G//an}bracketle{tz/an}bracketri}htis nonabelian and so ( G//an}bracketle{tz/an}bracketri}ht)′∼=C2 sinceG//an}bracketle{tz/an}bracketri}hthas three distinct abelian maximal subgroups. Thus |G′|= 4 and G′≤A= Φ(G). Taking h1∈H1−Aandh2∈H2−A, we have /an}bracketle{th1, h2/an}bracketri}ht=G and so s= [h1, h2]∈G′−/an}bracketle{tz/an}bracketri}ht. IfG′∼=C4, then zis a square in H1andH2and so both H1andH2are metacyclic (Proposition 1.1). Since d( H)≥3,Ghas exactly one nonmetacyclic maximal subgroup. But then Theor em 87.12 in [2] forn= 2 implies that Ghas an abelian maximal subgroup, a contradiction. Thus G′=/an}bracketle{ts, z/an}bracketri}ht∼=E4, where sis an involution. If s∈Z(G), then G//an}bracketle{ts/an}bracketri}ht would be abelian (because /an}bracketle{th1, h2/an}bracketri}ht=Gands= [h1, h2]), a contradiction. Hence s/ne}ati≀nslash∈Z(G) and so s/ne}ati≀nslash∈Z(H1) ors/ne}ati≀nslash∈Z(H2). Without loss of generality we may assume that s/ne}ati≀nslash∈Z(H1). Suppose that zis a square in H1, i.e., there is v∈H1such that v2=z. Suppose at the moment that v∈H1−Ain which case /an}bracketle{tv, G′/an}bracketri}ht=/an}bracketle{tv, s/an}bracketri}ht∼=D8 sincesv=sz. It follows that /an}bracketle{tv, G′/an}bracketri}ht=H1. Since C G(H1)≤H1(otherwise, d(G) = 3 ), Gwould be of maximal class (Proposition 1.7), a contradictio n (noting that 2-groups of maximal class have a cyclic maximal subgroup). Thus v∈A= Φ(G). In that case both H1andH2are metacyclic (Proposition 1.1) which together with d( H)≥3 allows us to use §87, part 20in [2]. If Ghas a normal elementary abelian subgroup of order 8, then we get g roups in part (a) of our theorem. If Ghas no normal elementary abelian subgroup of order 8, then we get the group of order 25given in part (b) of our theorem. Now we assume that zis not a square in H1which implies that H1is nonmetacyclic (Proposition 1.1). If h1is an involution, then sh1=szshows that /an}bracketle{th1, s/an}bracketri}ht∼=D8and so /an}bracketle{th1, s/an}bracketri}ht=H1is metacyclic, a contradiction. Hence o(h1) = 2n,n≥2. Set u=h2n−1 1so that u∈Z(H1) and u/ne}ati≀nslash∈G′sincezis not a square in H1andsh1=sz. We have E= Ω1(H1) =/an}bracketle{tz, s, u /an}bracketri}ht∼=E8,E/triangleleftequalG andE≤Awhich implies that H2is nonmetacyclic and therefore zis also not a square in H2. Since H1=E/an}bracketle{th1/an}bracketri}ht=/an}bracketle{th1, s/an}bracketri}ht, we have |H1|= 2n+2,|G|= 2n+3 andA=/an}bracketle{th2 1, s, z/an}bracketri}htis abelian of order 2n+1and type (2n−1,2,2). Also, H1is a splitting extension of G′by/an}bracketle{th1/an}bracketri}ht∼=C2n,n≥2. Since d( G/G′) = 2, we get thatG/G′is abelian of type (2n,2). We get G=FH1, where F∩H1=G′ and|F:G′|= 2 so that F=/an}bracketle{tG′, x/an}bracketri}htwith o( x)≤4. In fact, x2∈ /an}bracketle{tz/an}bracketri}ht. Indeed, if Fis not elementary abelian, then ℧1(F)∼=C2and℧1(F)≤G′. ButF/triangleleftequalGand so ℧1(F)≤Z(G) which implies that ℧1(F) = /an}bracketle{tz/an}bracketri}htsince G′/ne}ati≀nslash≤Z(G). Since /an}bracketle{txh1/an}bracketri}htG′/G′is another cyclic subgroup of index 2 in G/G′ (distinct from H1/G′),M=/an}bracketle{tG′, xh1/an}bracketri}htis a maximal subgroup of Gdistinct fromH1andM′=/an}bracketle{tz/an}bracketri}ht(since H′ 2=H′=/an}bracketle{tz/an}bracketri}ht). IfG′≤Z(M), then Mwould be abelian, a contradiction. We get sxh1=szand so M=/an}bracketle{txh1, s/an}bracketri}htis minimal nonabelian which gives M=H2. We may set h2=xh1, where [ h1, h2] =s 68 Z. BO ˇZIKOV AND Z. JANKO andG′=/an}bracketle{ts, z/an}bracketri}ht. From sxh1=szfollows sx= (sxh1)h−1 1= (sz)h−1 1=sh−1 1z= (sz)z=s and so Fis abelian. From s=
Question 16multiple-choice
In computational algebra and coding theory, reconstructing hidden structures often involves analyzing polynomial spaces over finite fields and leveraging properties of linear independence, sample spanning, and coset partitioning. Understanding bounds on subset sizes and the uniqueness of solutions is critical when solving equations and designing effective algorithms in these contexts.
In the setting of polynomials of degree at most \( p-1 \) in \( n \) variables over \( \mathbb{Z}_p \), which of the following statements accurately describes the upper bound on the ratio \(|R_k|/|V_k|\) for certain subspaces and associated sets, as established using coset partitioning arguments?
1) The ratio is always equal to \( 1 \) for all \( k \).
2) The ratio cannot exceed \( 1/2 \) for any \( k \).
3) The ratio is strictly less than \( 1/p \) for all \( k \).
4) The ratio is unbounded and can grow with \( n \).
5) The ratio is minimized when \( k = p-1 \).
6) The ratio is at most \( (p-1)/p \) for \( k = 1, \ldots, p-1 \).
7) The ratio equals \( (k-1)/k \) for all \( k \).
✓ Correct Answer:
The correct answer is 6) The ratio is at most \( (p-1)/p \) for \( k = 1, \ldots, p-1 \)..
📚 Reference Text:
the y(p−1)for the previously sampled y’s. This immediately implies that if our sample size is of the order of the dimension of Z(p−1) p[x1,...,x n], the span of y(p−1)for the samples yisZ(p−1) p[x1,...,x n] with high probability. In that case, the linear equations y(p−1)·U= 1 have exactly one solution, which is u∗. From this unique solution one can easily recover a vector vsuch that v=aufor some 0
Question 17multiple-choice
Quantum computing with qudits, which are quantum systems with more than two levels, enables algorithms to utilize higher-dimensional quantum registers. This approach can significantly impact resource requirements and error scaling for key algorithms such as phase estimation.
In the phase-estimation algorithm using qudit circuits, which advantage does increasing the qudit dimension d provide for quantum computing?
1) It allows the use of fewer controlled operations in the algorithm.
2) It eliminates the need for an inverse quantum Fourier transform.
3) It makes SWAP gates unnecessary for output reordering.
4) It enables phase estimation without requiring two registers.
5) It removes the requirement for eigenvector input states.
6) It exponentially reduces error rates and decreases the number of physical resources required for a given accuracy.
7) It prevents the need for quantum multiplexers during phase readout.
✓ Correct Answer:
The correct answer is 6) It exponentially reduces error rates and decreases the number of physical resources required for a given accuracy..
📚 Reference Text:
transform and ensure the correct sequence of j1j2/jn,as e r i e so fS W A P gates are applied at the end, which are not explicitly drawninFigure 10 . The QFT developed in qudit system offers a crucial subroutine for many quantum algorithm using qudits. Qudit QFT offerssuperior approximations where the magnitude of the errordecreases exponentially with dand the smaller error bounds are smaller [ 165]; which outperforms the binary case [ 34]. 3.2.2 Phase-Estimation Algorithm With Qudits With the qudit quantum Fourier transform, we are able to generalize the PEA to qudit circuits [ 26]. Similar to the PEA using qubit, the PEA in the qudit system is composed by tworegisters of qudits. The first register contains tqudits and t depends on the accuracy we want for the estimation. Weassume that we can perform a unitary operation Uto an arbitrary number of times using qudit gates and generate itseigenvector |u〉and store it using the second register ’s qudits [ 17]. We want to calculate the eigenvalue of |u〉where U|u〉/equalse 2πir|u〉 by estimating the phase factor r. The following derivations follow those in Ref. 26. For convenience, we rewrite the rational number ras r/equalsR/dt/equals⎥summationdisplayt k/equals0Rl/dl/equals0.R1R2/Rt. (100) As shown in Figure 11A , each qudit in the first register passes through the generalized Hadamard gate H≡F(d,d). For the lth qudit of the first register, we have F(d,d)|0l〉/equals1⎥radicaltpext⎥radicaltpext d√⎥summationdisplayd−1 kl/equals0|kl〉. (101) Then the lthqudit is used to control the operation Udt−lon the target qudits of the state |u〉in the second register, which gives CUdl−1|k〉⊗|u〉/equals|k〉⎥parenleftBigUdt−l⎥parenrightBigk|u〉/equalse2πikdt−lr|k〉⊗|u〉. (102) Note that the function of the controlled operation CUdt−lcan be considered as a “quantum multiplexer ”[24, 87, 139 ]. After executing all the controlled operations on the qudits, the quditsystem state turns out to be ⎛⎝⎥productdisplayt l/equals1⊗1⎥radicaltpext⎥radicaltpext d√⎥summationdisplayd−1 kl/equals0e2πikldt−lr|kl〉⎞⎠⊗|u〉. (103) Therefore, through a process called the “phase kick-back ”, the state of the first register receives the phase factor and becomes |Register 1 〉/equals1 dt/2⎥summationdisplaydt−1 k/equals0e2πirk|k〉. (104) The eigenvalue rwhich is represented by the state |R〉can be derived by applying the inverse QFT to the qudits in the first register: F−1⎥parenleftbigd,dt⎥parenrightbig|Register 1 〉/equals|R〉. (105) The whole process of PEA is shown in Figure 11B . To obtain the phase r/equalsR/dtexactly, we can measure the state of the first register in the computational basis. The PEA in qudit system provides a signi ficant improvement in the number of the required qudits and the error rate decreases Frontiers in Physics | www.frontiersin.org November 2020 | Volume 8 | Article 589504 14Wang et al. Qudits and High-Dimensional Quantum Computing exponentially as the qudit dimension increases [ 129]. A long list of PEA applications includes Shor ’s factorization algorithm [ 142]; simulation of quantum systems [ 1]; solving linear equations [ 69, 128]; and quantum counting [ 147]. To give some examples, a quantum simulator utilizing the PEA algorithm has been used to calculate the molecular ground-state energies [ 8] and to obtain the energy spectra of molecular systems [ 13, 41, 42, 84, 154 ]. Recently, a method to
Question 18multiple-choice
In mathematical physics, topological field theories connect algebraic structures like categories and algebras with the topology of manifolds, often leading to deep insights in representation theory and geometry. Categorical approaches to 2D and 4D topological field theories have advanced understanding of quantum invariants and dualities.
Which algebraic structure is essential for constructing category-valued 2D topological field theories via factorization homology, and also underlies the quantum symmetries present in braided tensor categories?
1) Commutative Hopf algebras
2) Symmetric monoidal categories
3) Modules for Drinfeld–Jimbo quantum groups
4) Derived categories of coherent sheaves
5) Real reductive Lie groups
6) Vertex operator algebras
7) Supercommutative algebras
✓ Correct Answer:
The correct answer is 3) Modules for Drinfeld–Jimbo quantum groups.
📚 Reference Text:
Title: Integrating quantum groups over surfaces Year: 2015 Paper ID: 4627ac6a237956c5297cc0091158a823996f4dc7 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/4627ac6a237956c5297cc0091158a823996f4dc7 Abstract: We apply the mechanism of factorization homology to construct and compute category‐valued two‐dimensional topological field theories associated to braided tensor categories, generalizing the (0,1,2)‐dimensional part of Crane–Yetter–Kauffman four‐dimensional TFTs associated to modular categories. Starting from modules for the Drinfeld–Jimbo quantum group Uq(g) we obtain in this way an aspect of topologically twisted four‐dimensional N=4 super Yang–Mills theory, the setting introduced by Kapustin–Witten for the geometric Langlands program.
Question 19multiple-choice
Modular tensor categories (MTCs) play a fundamental role in mathematical physics, especially in topological quantum field theory and quantum computing. The properties of modularity and unitarity, as well as specific fusion rules and the structure of the S-matrix, are central to their classification and physical applicability.
In the MTC of type Z(A1) at ℓ=5, which fusion rule characterizes the simple object X₁, and what key implication does this have for the category's use in topological quantum computing?
1) X₁ ⊗ X₁ = X₁ ⊕ 1₁ ⊕ X₁; the category is non-modular
2) X₁ ⊗ X₁ = 1₁; the category is abelian
3) X₁ ⊗ X₁ = 1₁ ⊕ X₁; the category supports non-abelian anyon models
4) X₁ ⊗ X₁ = X₁ ⊕ X₁; the category is not unitary
5) X₁ ⊗ X₁ = 0; the category is trivial
6) X₁ ⊗ X₁ = 1₁ ⊕ 1₁; the S-matrix is singular
7) X₁ ⊗ X₁ = 1₁ ⊕ X₁ ⊕ X₁; the category is non-unitary
✓ Correct Answer:
The correct answer is 3) X₁ ⊗ X₁ = 1₁ ⊕ X₁; the category supports non-abelian anyon models.
📚 Reference Text:
sub- category. We only give enough information to discuss the mod ularity and unitarity of the category. 5.1. Type Z(A1)atℓ= 5.The following MTC is obtained from C(sl2,5, eπi/5) by taking the subcategory of modules with integer highest we ights. There are two simple objects 1 1,and X1satisfying fusion rules: X1⊗X1= 1 1⊕X1and 1 1 ⊗Xi=Xi. TheS-matrix is S=/parenleftigg 11+√ 5 2 1+√ 5 2−1/parenrightigg and the twists: θ0= 1,θ1=e4πi/5. It is clear that det( S)/ne}ationslash= 0, and it follows from [ W] that the category is unitary (notice that the categorical dimensions are both positive). 5.2. Type B2at9th Roots of Unity. Consider the pre-modular categories C(so5,9, ejπi/9) with gcd(18 , j) = 1. There are 12 inequivalent isomorphism classes of simple objects. The simple iso-classes of objects are lab elled by ( λ1, λ2)∈1 2(N2) withλ1≥λ2. The twist coefficients for Xλisq/angbracketleftλ+2ρ,λ/angbracketrightwhere the form is twice the usual Euclidean form. The obstruction to modularity men tioned in the proof of Theorem 4.3 is labelled by γ:=1 2(5,5) The categorical dimension function is: dλ:=[2(λ1+λ2+ 2)][2( λ1−λ2+ 1)][2 λ1+ 3][2 λ2+ 1] [4][3][2][1]. One checks that the simple object Xγis indeed the cause of the singularity of the S-matrix, that is, Sγ,λ=dγdλfor all λ. Thus this category is not modular by Brugui` eres’ criterion, Theorem 4.1. Now let us consider the subcategory of C(so5,9, ejπi/9) with gcd(18 , j) = 1 generated by the simple objects labelled by integer weights : {(0,0),(1,0),(2,0),(1,1),(2,1),(2,2)}. The braiding and twists from the full category restrict, so t he entries of the S- matrix are computed from Formula (2.2). Taking the ordering of simple objects above, we denote the categorical dimensions by di0≤i≤5. The fusion matrix corresponding to (1 ,0) is: N1:= 0 1 0 0 0 0 1 0 1 1 0 0 0 1 0 0 1 0 0 1 0 1 1 0 0 0 1 1 1 1 0 0 0 0 1 1 . It is not hard to show that N1determines the other five fusion matrices by observing thatN1has six distinct eigenvalues and the fusion matrices commut e. There are a total of six categories corresponding to the six possible va lues of q. To describe the S-matrices we let αbe a primitive 18th root of unity, and set r1=−α−α2+α5, r2=α+α2−α4andr3=α4−α5. Then we get the following S-matrices (for the 14 ERIC C. ROWELL 6 choices of α): 1r2r31−1r1 r21 1 r1−r31 r31 1 r2−r11 1r1r21−1r3 −1−r3−r1−1 1 −r2 r11 1 r3−r21 . One checks that det( S)/ne}ationslash= 0 for any α, so these categories aremodular. A bit of Galois theory shows that there are only three distinct Sfor the six choices of α. Notice that it is already clear that the first column of Sis never positive, since both 1 and −1 appear regardless of the choice of α. So none of these categories is unitary. References [A] H.H. Andersen, Tensor products of quantized tilting modules , Comm. Math. Phys. 149(1991), 149-159. [AP] H.H. Andersen and J. Paradowski, Fusion categories arising from
Question 20multiple-choice
Group theory and computational complexity are deeply intertwined in modern mathematical physics, particularly in the study of fault-tolerant quantum computation using anyons and topological phases. Understanding the structure of groups and their representations is essential for analyzing both the hardness of computational problems and the robustness of quantum algorithms.
Which of the following statements accurately describes a property of simple groups that has implications for the existence of non-trivial homomorphisms in the context of free group evaluation homomorphisms?
1) Simple groups always have non-trivial normal subgroups, enabling trivial homomorphisms from free groups.
2) Any homomorphism from a free group to a simple group is necessarily trivial due to the group's structure.
3) The kernel of every evaluation homomorphism from a free group to a simple group is equal to the entire free group.
4) Simple groups cannot be used to construct normal subgroups via evaluation homomorphisms from free groups.
5) For simple groups, non-trivial evaluation homomorphisms from free groups must exist, implying there is always a word mapping non-trivially under some homomorphism.
6) All evaluation homomorphisms from free groups to simple groups must be isomorphisms.
7) Simple groups have abelian structure, guaranteeing the normality of all their subgroups.
✓ Correct Answer:
The correct answer is 5) For simple groups, non-trivial evaluation homomorphisms from free groups must exist, implying there is always a word mapping non-trivially under some homomorphism..
📚 Reference Text:
have |{φ∈H(K(Kℓ c);C,G)|ψ(xi) =ci}|≥ 2ℓ|{φ∈H(K(Kℓ c);C,G)→C|ψ(xi) =di}|. (Observe that, as presented, this is only efficient if G=Anhas constant size.) Lemma 6. Consider the alternating group on msymbolsAmand consider any conjugacy class with at least 4fixed points, say, m−3,m−2,m−1andm. Since the conjugacy class is non-trivial, it has a kcycle withk≥2. Pickαin this class to have (1 2... k)in thatkcycle. Now, consider the equation y=α(1k)(m−2m−3)y. This equation has at least two solutions. Proof. It can easily be checked that two solutions are given by eleme nts that differ from αonly in the k cycle. The first one contains (1 2 ... k−1m) and the second (1 2 ... k−1m−1) in thatkcycle. Lemma 7. LetGbe a finite group and Fkbe the free group on kgenerators{x1,...,x k}. For an element d= (d1,...,d k)∈Gk, letφdbe the evaluation homomorphism that carries xitodiand define Ad= kerφd={w∈Fk|φd(w) = 1}. For two elements c,d∈Gk, defineCto be the subgroup generated by the ci,Dto be the subgroup generated by thedi, and Ad(c) ={φc(w)|w∈Ad}. Then 1.Ad(c)is normal in C. In particular, when the cigenerateG,Ad(c)is normal in G. 2. Ifcanddare not related by a homomorphism of G(which is to say that no homomorphism from Dto Ccarries each citodi) thenAd(c)/\e}ati≀\slash= 1. 21 Proof. For part 1, ker(φd) is certainly normal in Fksinceφdis a group homomorphism. Let Cbe the subgroup of Ggenerated by the ci, andg∈C; we wish to show that g−1Ad(c)g=Ad(c). Ifw∈Fkis a word in the free group for which φc(w) =g, we havew−1Adw=Adand hence that g−1Ad(c)g=φc(w−1Adw) =φc(Ad) =Ad(c), as desired. As for part 2, letCdenote the subgroup generated by {ci}andDthe subgroup generated by {di}. Observe that if Ad(c) is trivial we have ker φd⊂kerφc. In this case, the natural quotient map qc:D∼=F/kerφd→C∼=F/kerφc yields a homomorphism ψ:D→Cfor whichψ:di/ma√st≀→ci. To be precise, let qcdenote the quotient map qc:F/kerφd→F/kerφcand letiddenote the inverse of the isomorphism φdinduces from F/kerφdtoD; observe that id(di) =xi(kerφd). Then the map ψ=φc◦qc◦idis a homomorphism of DontoCthat carries eachdiontoci; see Figure 6. D F/kerφd F/kerφc C idφd qc φc Figure 6: The homomorphism ψ:di/ma√st≀→ci. The above lemma implies that if Gis simple then H1=G(H1cannot be trivial because canddare not related by an automorphism). This means that there exists a w ordwin variables xisuch thatw(c) =zand w(d) = 1. Proof of Theorem 4.The first two follow from the definitions of BPP andSBP and the following lemmas. The # P-completeness follows from the fact that exact evaluation o f the success probability of a randomized computation is # P-complete. 6 Quantum computation with anyons Anyons are particles which exist in two dimensions and have e xotic statistics. Anyons are useful for quantum computation because quantum information can be stored on a s ystem of anyons in a non-local fashion. This means that local errors do not corrupt the quantum informati on and a computer based on anyons will be inherently fault tolerant to local errors. For a tutorial on quantum computation using anyons, see the notes by Preskill [ 30]. In this section, we give short introduction to anyons desc ribed by D(G) for a
Question 21multiple-choice
Quaternions are four-dimensional mathematical entities that have played significant roles in physics, mathematics, and modern computational fields such as computer graphics and robotics. Their unique algebraic properties have influenced the development of group theory and the understanding of spatial rotations.
Which of the following properties distinguishes the quaternion group Q8 from the cyclic group Z4 in the context of group theory and symmetry operations?
1) Q8 is abelian while Z4 is non-abelian
2) Q8 has non-commutative (non-abelian) multiplication, whereas Z4 is abelian
3) Q8 contains only elements of order 4, while Z4 contains elements of order 8
4) Z4 has elements corresponding to the imaginary units i, j, k, while Q8 does not
5) Q8 and Z4 both have subgroup structures identical to the symmetric group S4
6) Q8 is a cyclic group, while Z4 is not
7) Z4 represents rotations in three dimensions, while Q8 cannot be used to encode rotations
✓ Correct Answer:
The correct answer is 2) Q8 has non-commutative (non-abelian) multiplication, whereas Z4 is abelian.
📚 Reference Text:
have three dimensions...The mathematical quaternion partakes of both these elements; in technical language it may be said to be “time plus space”, or “space plus time”: and in this sense it has, or at least involves a reference to, four dimensions. And how the One of Time, for Space the Three, Might in the Chain of Symbols girdled be. Hamilton was the first to use the terms “scalar” and “vector” in their modern sense when he treated quaternions as having scalar (real) and vector (imaginary) part; he referred to quaternions with zero scalar part as pure quaternions. Even though many physicists, including James Clerk Maxwell, used quaternions with en- thusiasm, they were eventually discarded in favor of the vector algebra promoted by William Gibbs and Olivier Heaviside. One of the reasons was that in most physical applications only the imagi- nary part of the quaternions was required, so they came to be viewed as unnecessarily complicated. In essence, the vector algebraists simply extracted the vector part of the quaternions, discarding the scalar part, and developed a more consistent notation. But the drawbacks may have outweighed the benefits. It has been argued that if quaternions “had been used in the correct way from early on, they would have pre-empted many later physical developments that came about much more tortuously” [28]. 42 A comment made by Bertrand Russell on Plato’s Theory of Ideas [29] comes to mind: Any attempt to divide the world into portions, of which one is more ’real’ than the other, is doomed to failure. The heated controversy over the usefulness of quaternions continued for several generations after Hamilton’s death. 4.1.3 The Revival The impact of quaternions on the development of mathematical thought is undeniable: much of the work in the theory of algebras descended from Hamilton’s discovery. That quaternions arise naturally in standard quantum mechanics when time reversals are considered was first noted by Dyson [31] in 1962. Since then, other physicists have formulated quaternionic generalizations of the postulates of quantum mechanics, quantum field theory, and gravitational theory. Advantages of quaternionic forms have been exploited for attitude control in aeronautics and astronautics [34]. Most recently, quaternions have found extensive applications in the fields of 3D computer graphics, animations, and virtual reality; and it all began with a seminal 1985 paper by Shoemake [39]. The Euler-angle method for implementing 3D rotations is susceptible to Gimbal lock, where the order of rotations causes the failure of rotation to appear as expected. Quaternions provide a solution to the problem, allowing the programmer to rotate an object through an arbi- trary rotation axis and angle instead of a series of successive rotations. The errors accumulating due to matrix multiplication are avoided, as quaternions representing the axii of rotations are mul- tiplied instead. Moreover, quaternions can be interpolated, thus producing smooth and predictable rotation effects. 43 4.2 The Group Q4 The generalized quaternion group of order 2n,n3, is given by the presentation Q2n=hx;yjx2n 1=1;x2n 2=y2;y 1xy=x 1i (4.17) Forn=2, Q4=Z4=hx;yjx2=1;x=y2i=hxjx4=1i (4.18) The only subgroups of Z4are H1=f0g=h4i
Question 22multiple-choice
Quantum algorithms for the hidden subgroup problem (HSP) play a crucial role in computational group theory and cryptography. Efficient solutions often depend on the structure of the groups involved, influencing both algorithm design and cryptographic security.
Which of the following group-theoretical properties allows certain instances of the non-Abelian hidden subgroup problem to be solved efficiently with quantum algorithms?
1) Having a simple group structure
2) Containing an infinite cyclic subgroup
3) Possessing a large center
4) Being nilpotent of high class
5) Admitting a non-normal subgroup of small order
6) Exhibiting a non-trivial direct product decomposition
7) Having an elementary Abelian normal 2-subgroup of small index
✓ Correct Answer:
The correct answer is 7) Having an elementary Abelian normal 2-subgroup of small index.
📚 Reference Text:
arXiv:quant-ph/0102014v1 2 Feb 2001Efficient quantum algorithms for some instances of the non-Ab elian hidden subgroup problem∗ G´ abor Ivanyos†Fr´ ed´ eric Magniez‡Miklos Santha§ October 15, 2018 Abstract Inthispaperweshowthatcertainspecialcasesofthehidden subg roupproblemcanbe solved in polynomial time by a quantum algorithm. These special cases involve finding hidden normal subgroups of solvable groups and permutation groups, finding hidd en subgroups of groups with small commutator subgroup and of groups admitting an elementary Abelian normal 2-subgroup of small index or with cyclic factor group. 1 Introduction A growing trend in recent years in quantum computing is to cas t quantum algorithms in a group theoretical setting. Group theory provides a unifying fram ework for several quantum algorithms, clarifies their key ingredients, and therefore contributes to a better understanding why they can, in some context, be more efficient than the best known classica l ones. The most important unifying problem of group theory for the p urpose of quantum algorithms turned out to be the hidden subgroup problem (HSP) which can be cast in the following broad terms. Let Gbe a finite group (given by generators), and let Hbe a subgroup of G. We are given (by an oracle) a function fmapping Ginto a finite set such that fis constant and distinct on different left cosets of H, and our task is to determine the unknown subgroup H. While no classical algorithm is known to solve this problem i n time faster than polynomial in the order of the group, the biggest success of quantum comp uting until now is that it can be solved by a quantum algorithm efficiently , which means in time polynomial in the logarithm of the order of G,whenever the group is Abelian. The main tool for this solutio n is the (approximate) quantum Fourier transform which can be efficiently implement ed by a quantum algorithm [17]. Simon’s algorithm for finding an xor-mask [26], Shor’s semin al factorization and discrete logarithm finding algorithms [25], Boneh and Lipton’s algorithm for fin ding hidden linear functions [6] are all special cases of this general solution, as well as the alg orithm of Kitaev [17] for the Abelian stabilizer problem, which was the first problem set in a gener al group theoretical framework. That all these problems are special cases of the HSP, and that an effi cient solution comes easily once an efficient Fourier transform is at our disposal, was realize d and formalized by several people, ∗Research partially supported by the EU 5th framework progra ms QAIP IST-1999-11234, and RAND-APX, IST- 1999-14036, by OTKA Grant No. 30132, and by an NWO-OTKA grant . †Computer and Automation Institute, Hungarian Academy of Sc iences, L´ agym´ anyosi u. 11., H-1111 Budapest, Hungary, e-mail: Gabor.Ivanyos@sztaki.hu . ‡CNRS–LRI, UMR 8623 Universit´ e Paris–Sud, 91405 Orsay, Fra nce, e-mail: magniez@lri.fr . §CNRS–LRI, UMR 8623 Universit´ e Paris–Sud, 91405 Orsay, Fra nce, e-mail: santha@lri.fr . 1 including Brassard and Høyer [7], Mosca and Ekert [22] and Jo zsa [15]. An excellent description of the general solution can be
Question 23multiple-choice
Quantum computing exploits unique principles of quantum mechanics, such as superposition and entanglement, to tackle computational problems that are intractable for classical computers. Its theoretical foundation includes important concepts in quantum information science and has profound implications for cryptography and algorithm design.
Which theorem fundamentally asserts that it is impossible to produce an exact copy of an arbitrary unknown quantum state, shaping the security and limitations of quantum information transmission?
1) Bell’s theorem
2) No-cloning theorem
3) Superposition principle
4) Church-Turing thesis
5) Quantum entanglement theorem
6) Grover’s algorithm
7) Unitarity principle
Categorification in knot theory uses advanced algebraic structures, such as foams and higher representation theory, to generalize classical invariants and explore deep links between topology and quantum groups. Modified foam categories are important for improving functorial properties and robustness of categorified knot invariants.
Which modification in the foam category for sl(2) is crucial for achieving better functoriality in Khovanov homology, and what analogous approach is suggested for sl(3) categories?
1) Blanchet’s modified foams, with analogous modifications proposed for sl(3) categories
2) Drinfeld double construction, with dual versions for sl(3)
3) Witten’s topological twists, generalized for sl(3)
4) MOY calculus enhancements, extended to sl(3) settings
5) Colored Jones polynomial extensions, applied to sl(3) algebra
6) Use of matrix factorizations, adapted for sl(3) foams
7) Temperley-Lieb algebra modifications, transferred to sl(3) context
✓ Correct Answer:
The correct answer is 1) Blanchet’s modified foams, with analogous modifications proposed for sl(3) categories.
📚 Reference Text:
Title: Khovanov homology is a skew Howe 2-representation of categorified quantum sl(m) Year: 2012 Paper ID: c5a24f2b3390d4ee2da8737782a092ea4cbb0041 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/c5a24f2b3390d4ee2da8737782a092ea4cbb0041 Abstract: We show that Khovanov homology (and its sl(3) variant) can be understood in the context of higher representation theory. Specifically, we show that the combinatorially defined foam constructions of these theories arise as a family of 2-representations of categorified quantum sl(m) via categorical skew Howe duality. Utilizing Cautis-Rozansky categorified clasps we also obtain a unified construction of foam-based categorifications of Jones-Wenzl projectors and their sl(3) analogs purely from the higher representation theory of categorified quantum groups. In the sl(2) case, this work reveals the importance of a modified class of foams introduced by Christian Blanchet which in turn suggest a similar modified version of the sl(3) foam category introduced here.
Question 25multiple-choice
Quantum cryptanalysis explores how quantum algorithms and memory models can accelerate attacks on lattice-based cryptosystems, including those built upon the Learning with Errors (LWE) problem. Techniques such as quantum random access classical memory (QRACM), quantum amplitude estimation, and the Quantum Fourier Transform (QFT) are central to these advancements.
Which of the following improvements is enabled when quantum-augmented dual lattice attacks on LWE employ quantum random access classical memory (QRACM) with unit cost?
1) Reduction in the modulus size required for secure cryptosystems
2) Elimination of the need for discrete Gaussian sampling
3) Exponential speedup in classical memory usage
4) Replacement of FFT with Grover search for coefficient identification
5) Ability to solve all lattice problems in polynomial time
6) Quantum speedup in searching FFT coefficients above a threshold by leveraging sparsity and amplitude estimation
7) Guarantee of security for all post-quantum cryptographic schemes
✓ Correct Answer:
The correct answer is 6) Quantum speedup in searching FFT coefficients above a threshold by leveraging sparsity and amplitude estimation.
📚 Reference Text:
Title: Quantum Augmented Dual Attack Year: 2022 Paper ID: d01adf744d3e5e28dbab2a5b868495143bbcb77b Source: semantic-scholar URL: https://www.semanticscholar.org/paper/d01adf744d3e5e28dbab2a5b868495143bbcb77b Abstract: We present a quantum augmented variant of the dual lattice attack on the Learning with Errors (LWE) problem, using classical memory with quantum random access (QRACM). Applying our results to lattice parameters from the literature, we find that our algorithm outperforms previous algorithms, assuming unit cost access to a QRACM. On a technical level, we show how to obtain a quantum speedup on the search for Fast Fourier Transform (FFT) coefficients above a given threshold by leveraging the relative sparseness of the FFT and using quantum amplitude estimation. We also discuss the applicability of the Quantum Fourier Transform in this context. Furthermore, we give a more rigorous analysis of the classical and quantum expected complexity of guessing part of the secret vector where coefficients follow a discrete Gaussian (mod \(q\)).
Question 26multiple-choice
In quantum topology and mathematical physics, projective and linear representations of the mapping class group play a crucial role in understanding topological quantum field theories (TQFTs) and their associated quantum symmetries. The interplay between group cohomology, roots of unity, and geometric constructions provides subtle distinctions between different representation types.
Under what condition does the projective representation of the mapping class group arising from an abelian TQFT associated with a q-deformation of U(1) become linearizable?
1) When q is a primitive even root of unity
2) When the underlying surface is non-orientable
3) When the mapping class group is abelian
4) When the Heisenberg group is replaced by a finite cyclic group
5) When q is an odd root of unity
6) When Lagrangian correspondences are not used in the construction
7) When the Schrödinger representation is infinite-dimensional
✓ Correct Answer:
The correct answer is 5) When q is an odd root of unity.
📚 Reference Text:
Title: Abelian TQFTS and Schrödinger local systems Year: 2023 Paper ID: 6265d84115c91c73711e5bd41edcb2135d798178 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/6265d84115c91c73711e5bd41edcb2135d798178 Abstract: In this paper, we construct an action of 3-cobordisms on the finite dimensional Schrödinger representations of the Heisenberg group by Lagrangian correspondences. In addition, we review the construction of the abelian Topological Quantum Field Theory (TQFT) associated with a q -deformation of U(1) for any root of unity q . We prove that, for 3-cobordisms compatible with Lagrangian correspondences, there is a normalization of the associated Schrödinger bimodule action that reproduces the abelian TQFT.The full abelian TQFT provides a projective representation of the mapping class group \mathrm{Mod}(\Sigma) on the Schrödinger representation, which is linearizable at odd root of 1. Motivated by homology of surface configurations with Schrödinger representation as local coefficients, we define another projective action of \mathrm{Mod}(\Sigma) on Schrödinger representations. We show that the latter is not linearizable by identifying the associated 2-cocycle.
Question 27multiple-choice
In the representation theory of symmetric groups, Young tableaux, tabloids, and Specht modules are fundamental concepts used to construct and understand irreducible representations. The interplay between group actions on tableaux and algebraic structures underlies important results in both mathematics and quantum physics.
Which statement correctly characterizes the algebraic behavior of swap relations in irreducible representations associated with two-row Young tableaux, specifically those of the form λ = [n−k, k]?
1) The swap relations vanish under evaluation in these representations, reflecting a simpler symmetry structure.
2) The swap relations generate nontrivial constraints that prevent the irreducibility of Specht modules for two-row tableaux.
3) The swap relations always result in incompatible algebraic structures, regardless of the tableau shape.
4) The swap relations lead to the formation of additional rows in the corresponding Young diagrams.
5) The swap relations have coefficients that depend only on the parity of the number of columns.
6) The swap relations in two-row tableaux produce polytabloids with alternating signs in every entry.
7) The swap relations correspond to the action of the Quantum Monte Carlo Hamiltonian on multipartite irreducible representations.
✓ Correct Answer:
The correct answer is 1) The swap relations vanish under evaluation in these representations, reflecting a simpler symmetry structure..
📚 Reference Text:
tableau of shapeλ⊢nis a filling of the Young di- agramλby integers 1 ,2,...,n such that each box is assigned a unique integer. The action of the groupSnon a Young tableau tfollows by letting π∈Snact on the entries of t. Permut- ing entries within each row of a tableau tgives us an equivalence class of tableaux – a tabloid {t1,t2,...,tk}:={t}. (3.5) Accepted in Quantum2024-03-25, click title to verify. Published under CC-BY 4.0. 22 The group action extends to tabloids as π{t}={π(t1),π(t2),...,π (tk)}. The so-called Specht irreducible module VλofC[Sn] is spanned by polytabloids et=/summationdisplay π∈Ctsgn (π)π{t} (3.6) wheretranges over Young tableau of shape λ, andCtis the set of permutations that permute elements only within the columns of t. For a 3-row irrep given by λ= [λ1,λ2,λ3], letTbe the Young tableau T=1 2··· ··· λ1 λ1+1... ... λ 2 λ2+1... λ 3.(3.7) andeTthe corresponding polytabloid. Consider the transpositions (1 λ1+ 1), (1λ2+ 1), (λ1+ 1λ2+ 1) all of which permute elements within the first column of Tand are thus contained in CT. Note that (1λ1)eT=e(1λ1)T (3.8) and on expanding the right-hand side of Eq. (3.8) we see that Tappears with coefficient sgn ((1λ1+ 1)) =−1. By the exact same reasoning, the coefficients of Tin (1λ2)eTand (λ1λ2)eTalso equal−1. Now, each of (1 λ1+ 1)(λ1+ 1λ2+ 1) = (1λ2+ 1λl+ 1) and (λ1+ 1λ2+ 1)(1λ1+ 1) = (1λ1+ 1λ2+ 1) is an even permutation contained in CT. Repeating the above for (1 λ1+ 1)(λ1+ 1λ2+ 1)eTand (λ1+ 1λ2+ 1)(1λ1+ 1)eT, we see that the coefficient of Tin these must be +1. It follows that [(1λ1+ 1) + (1λ2+ 1) + (λ1+ 1λ2+ 1)−1]eT̸={(1λ1+ 1),(λ1+ 1λ2+ 1)}eT(3.9) in the irrep specified by λ, and more generally in any irrep with 3 or more rows. That is, sijsjk+sjksij−(sij+sjk+sik−1) does not vanish under the evaluation sij=ρλ(i j) for the above irrep λ. Thus we have proved that the swap relations are incompatible with a ≥3 row Young Tableaux. For the converse, consider distinct indices i,j,k and a tabloid Tof shape [n−k,k]. Letπ∈CT. We distinguish two cases: (a)i,j,k lie in the same row of π{T}. Then each of the terms sij,sjk,sikacts as the identity on π{T}, whence /parenleftbigsijsjk+sjksij−(sij+sjk+sik−1)/parenrightbig(π{T}) = 0. (3.10) (b)i,jlie in the same row of π{T}, butkdoes not. Then, sij(π{T}) =π{T}, sijsjk(π{T}) =sik(π{T}), sjksij(π{T}) =sjk(π{T}). As before, this implies Eq. (3.10). This shows that for each π∈CT, Eq. (3.10) holds, whence sijsjk+sjksij−(sij+sjk+ sik−1) vanishes under the evaluation sij=ρλ(i j) for any irrep λ= [n−k,k]. Accepted in Quantum2024-03-25, click title to verify. Published under CC-BY 4.0. 23 Theorem 3.6 ([Pro07 , Theorem, §6.1]).The algebras Aswap n andMswap n are isomorphic. More precisely, the natural map Aswap n→Mswap n, sij∝⇕⊣√∫⊔≀→Swapij is an isomorphism. Proof. An immediate corollary of Proposition 3.5. The isomorphism between the Symbolic Swap Algebra and Swap Matrix Algebra given above lets us identify the QMC Hamiltonian associated with a graph Gwith an element of the Symbolic Swap Algebra. We make this clear in the following definition. Definition 3.7. For any graph gdefine the symbolic QMC Hamiltonian
Question 28multiple-choice
In the study of abelian p-groups, the structure of automorphism groups and characteristic subgroups are crucial for understanding group extensions and classifications. Properties such as divisibility, direct decompositions, and the behavior of homocyclic components impact the nature of possible group extensions.
Which statement correctly describes the necessary and sufficient condition for the automorphism group Coo to equal Aut A in an abelian p-group A with divisible part D and reduced part R?
1) Coo = Aut A if and only if Cpoo is contained in Aut D or in Aut R.
2) Coo = Aut A only if Cpoo is a non-central subgroup of A.
3) Coo = Aut A if and only if every extension of A by C^oo is non-nilpotent.
4) Coo = Aut A provided that D is reduced and R is divisible.
5) Coo = Aut A if and only if Cpoo equals Aut(A/C) for all characteristic subgroups C.
6) Coo = Aut A if and only if every homomorphism from Coo to Aut A is surjective.
7) Coo = Aut A only if A has no homocyclic components.
✓ Correct Answer:
The correct answer is 1) Coo = Aut A if and only if Cpoo is contained in Aut D or in Aut R..
📚 Reference Text:
if Coo = Aut A, the corresponding split extension would be a nilpotent /?-group containing a non-central Cpoo, which is impossible. Conversely, if (ii) holds then every homomorphism from Coo to Aut A is trivial so that every extension G of A by C^oo is a central extension and thus nilpotent. In fact it is abelian since G has locally cyclic central factor-group. The equivalence of (iii) and (iv) is immediate since D and every homocyclic factor of L are direct factors of A. That (ii) implies (iii) is a direct consequence of (2.8). The hard part of this theorem is the proof that (iii) implies (ii). We need some lemmas to enable us to do it. LEMMA 3.4.1. Let C be a characteristic subgroup of an abelian group A and suppose that AI C is a p-group. //Coo = Aut A, then either Cpoo ^ Aut C or Cpoo ^ Aut(v4/C). https://doi.org/10.4153/CJM-1986-050-9 Published online by Cambridge University Press 1030 J. BUCKLEY AND J. WIEGOLD Proof. Every automorphism of A induces one on C and one on A/C. Furthermore, the obvious map 0:Aut A -> Aut C X Aut(A/C) has kernel the stability group of the chain 1 â C S A, which is isomorphic to Hom(A/C, C). Since A/C is a /?-group it is easily shown that Hom(v4/C, C) has no elements of infinites-height and so ker 0 contains no Coo. Thus Cp0o â im 6 ^ Aut C X Aut(,4/C). But any Coo in the direct product Aut A X Aut(,4/C) cannot be in the kernels of both projections, so Cpoo g Aut A or C^oo ê A\xt(A/C). COROLLARY. Let A be an abelian p-group with divisible part Z), so that A = D X R where R is reduced. Then Cœ = Aut A if and only if Cpoo â Aut D or Cpoo g Aut R. LEMMA 3.4.2. Let A be an abelian p-group, L a basic subgroup of A, and let Ln be the homocyclic component ofL of exponent p n. Then for each «èl, Z>|Z>2 • • • LnÂP is characteristic in A. Proof. Recall that A = L\L 2 . . . L„A*, where A* = AP"L* and L* = (ULjm > n). We shall show that (L XL2 . . . L n)a is contained in LXL2 . . . LnAp for each n ^ 1 and each a e Aut A. For each JC G LXL2 . . . L tv x = x xyxay , where xx G LjL2 . . . L, 7, _yj G L* and a & A. Since x /;" = x px = 1 it follows that (y]apY = I- So ><' = a~p2n and jf G L*. But L* is a direct factor of the pure subgroup L and so it is itself pure in A. Thus yP\ = yPi> where _y2 G L*. https://doi.org/10.4153/CJM-1986-050-9 Published online by Cambridge University Press NILPOTENT EXTENSIONS 103 1 Since (y^2P"f = i, it follows that y\yïP is
Question 29multiple-choice
In the study of moduli spaces arising from group actions on vector spaces, the non-emptiness and dimension of quotient spaces can be determined using recursive operations and stability functions. The structure and properties of these spaces often depend on integer parameters and explicit combinatorial loci.
When analyzing the non-emptiness of quotient spaces parameterized by triples (A, b, c) with A ≥ 3, which of the following best describes the set SA of pairs (b, c) that must be checked for non-emptiness?
1) SA consists of all pairs (b, c) with b and c both greater than A
2) SA is infinite and contains all pairs (b, c) where bc > A
3) SA is the set of pairs (b, c) with b = c and both less than or equal to A
4) SA is a finite region near the curve bc = A, symmetric about b = c and contained in b < A, c < A
5) SA is the set of pairs (b, c) with b and c both equal to 1
6) SA contains all pairs (b, c) such that b divides c exactly
7) SA is the collection of all pairs (b, c) with b or c equal to zero
✓ Correct Answer:
The correct answer is 4) SA is a finite region near the curve bc = A, symmetric about b = c and contained in b < A, c < A.
📚 Reference Text:
and that the quotient corresponding to (b,c,A)is nonempty. Proof.By part (3) of Proposition 5.3, ∆ <0 implies ∆ ≤ −2; this is also implied by d3≥d∗(d1,d2) from the definition of d∗. Thus, suppose that the quotient is non-empty for dimensions ( A,d2,d3) with ∆(A,d2,d3)≤ −2. Consider applying the operation γ: (A,B,C)→(A,AB−C,B) (5.14) repeatedly without reordering todefineasequence (( A,B0,C0)≡(A,d2,d3),(A,B1,C1),...). Since we must have C0≥A0B0/2 for ∆<0, the initial step does not increase the sum of the elements. Further, all elements remain positive unless we end up on a triple for which Ci=ABi. Thus, repeating the operation γmust either bring us to a triple ( A,b,Ab) or to a triple (A,b,c) for which the operation γdoes not decrease the sum of the elements. In the latter case, we can show that bandcsatisfy b≤A 2c c≤A 2b bc≥A Abc −A2−b2−c2+4≤0. (5.15) The first pair of inequalities for ccome from demanding that γapplied to ( A,b,c) does not decrease the sum and that ( A,b,c) came by acting with γon a triple with a sum that was 37 not smaller. The third inequality follows since we are assuming the quot ient is not empty. The fourth inequality is the statement that ∆ ≤ −2. If (A,B,C) descends via γto (A,b,c), the quotient corresponding to this triple must be non- empty, sothepair( b,c)isintheset SAdefinedbytheproposition. Thus, anytriple( A,d2,d3) with ∆≤ −2 descends via the recursion step to ( A,k,Ak) or (A,b,c) with (b,c)∈SA. To determine the form of such triples ( A,d2,d3) explicitly, note that the inverse of the operation γis the operation Bi+1=CiCi+1=ACi−Bi (5.16) Combining these we find that BiandCimust be successive terms in the sequence fi+1=Afi−fi−1, (5.17) where the allowed starting values are ( f0,f1) = (k,Ak) or (f0,f1)∈SA. Remark 5.5. The general terms in the sequence defined by (5.12) can be given ex plicitly via a generating function as fi=f0+(f1−Af0)x 1−Ax+x2|xn. (5.18) ForA≥3, the region defined by (5.13) covers a narrow band of the plane ne ar the curve bc=A, symmetric about the line b=cand contained in the region b < A,c < A. Thus, while the definition of SAstill involves the recursion from Theorem 5.1, we need only check a finite number of points (on the order of A) in the region (5.13) to determine the set, after which we can use Proposition 5.4 to give an explicit formula for all ∆ <0 triples (A,d2,d2) with a nonempty quotient. As examples, we have S3= (3,2),(2,2),(2,3) S4= (4,2),(3,2),(2,3),(2,4) S5= (5,2),(4,2),(2,4),(2,5) (5.19) We consider the special case A= 2 presently. Example 5.6. For dimensions (2 ,d2,d3) the quotient P(H)//Gis non-empty if and only if (d1,d2,d3) = (2,b,b) forb≥2 or (2,kb,(k+1)b) for positive integers k,bwithkb>1. The quotient is a single point except for ( d1,d2,d3) = (2,b,b) withb >3, in which case it has dimensionb−3. Proof.For this case, the range dn−1≤dn< d∗is empty, so the only non-empty quotients are those covered by Proposition 5.4. For A= 2, the definition (5.12) of the sequence in Proposition 5.4 may be written as fi+1−fi=fi−fi−1so the sequence is arithmetic.
Question 30multiple-choice
In computational group theory, the complexity of decision problems often depends on the structural properties of the groups involved. Extensions of classical problems such as the Post correspondence problem to group settings reveal deep connections between group structure and algorithmic tractability.
Which of the following statements accurately characterizes the computational complexity of the bounded Post correspondence problem (PCP) in non-elementary hyperbolic groups?
1) It is solvable in logarithmic space.
2) It is undecidable in all cases.
3) It is NP-complete.
4) It is polynomial-time solvable for arbitrary input size.
5) It is co-NP-hard but not NP-hard.
6) It is decidable only for abelian groups.
7) It is PSPACE-complete.
✓ Correct Answer:
The correct answer is 3) It is NP-complete..
📚 Reference Text:
Title: The Post correspondence problem in groups Year: 2013 Paper ID: b2e2c40e98b47cb677c20902e6c929c3385eb6f1 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/b2e2c40e98b47cb677c20902e6c929c3385eb6f1 Abstract: Abstract We generalize the classical Post correspondence problem (𝐏𝐂𝐏n) and its non-homogeneous variation (𝐍𝐏𝐂𝐏n) to non-commutative groups and study the computational complexity of these new problems. We observe that 𝐏𝐂𝐏n is closely related to the equalizer problem in groups, while 𝐍𝐏𝐂𝐏n is connected to the double twisted conjugacy problem for endomorphisms. Furthermore, it is shown that one of the strongest forms of the word problem in a group G (we call it the hereditary word problem) can be reduced to 𝐍𝐏𝐂𝐏n in G in polynomial time. The main results are that 𝐏𝐂𝐏n is decidable in a finitely generated nilpotent group in polynomial time, while 𝐍𝐏𝐂𝐏n is undecidable in any group containing free non-abelian subgroups (though the argument is very different from the classical case of free semigroups). We show that the double endomorphism twisted conjugacy problem is undecidable in free groups of sufficiently large finite rank. We also consider the bounded 𝐏𝐂𝐏 and observe that it is in 𝐍𝐏 for any group with 𝐏-time decidable word problem, meanwhile it is 𝐍𝐏-hard in any group containing free non-abelian subgroups. In particular, the bounded 𝐏𝐂𝐏 is 𝐍𝐏-complete in non-elementary hyperbolic groups and non-abelian right angle Artin groups.
Question 31multiple-choice
Hardware accelerators play a vital role in enabling efficient processing for both digital signal processing and cryptographic applications. Unified architectures that support multiple computational transforms can optimize resources for modern embedded systems.
Which architectural feature is essential for a hardware accelerator to efficiently support both Fast Fourier Transform (FFT) for signal processing and Number Theoretic Transform (NTT) for lattice-based cryptography?
1) Implementation of high-speed floating-point multipliers
2) Addition of modular reduction circuitry and control logic modifications
3) Use of analog-to-digital converters for input transformation
4) Dedicated complex domain arithmetic units for NTT operations
5) Inclusion of error correction modules for cryptographic protocols
6) Separate data paths for FFT and NTT computation
7) Exclusive reliance on silicon area minimization techniques
✓ Correct Answer:
The correct answer is 2) Addition of modular reduction circuitry and control logic modifications.
📚 Reference Text:
Title: A Unified Hardware Accelerator for Fast Fourier Transform and Number Theoretic Transform Year: 2025 Paper ID: c04e27334a322c4482040993ca7eea33dc44cbf3 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/c04e27334a322c4482040993ca7eea33dc44cbf3 Abstract: The Number Theoretic Transform (NTT) is an indispensable tool for computing efficient polynomial multiplications in post-quantum lattice-based cryptography. It has strong resemblance with the Fast Fourier Transform (FFT), which is the most widely used algorithm in digital signal processing. In this work, we demonstrate a unified hardware accelerator supporting both 512-point complex FFT as well as 256-point NTT for the recently standardized NIST post-quantum key encapsulation and digital signature algorithms ML-KEM and MLDSA respectively. Our proposed architecture effectively utilizes the arithmetic circuitry required for complex FFT, and the only additional circuits required are for modular reduction along with modifications in the control logic. Our implementation achieves performance comparable to state-of-the-art ML-KEM / ML-DSA NTT accelerators on FPGA, thus demonstrating how an FFT accelerator can be augmented to support NTT and the unified hardware can be used for both digital signal processing and post-quantum lattice-based cryptography applications.
Question 32multiple-choice
In the study of group theory and representation theory, the Modular Isomorphism Problem investigates the relationship between the structure of finite p-groups and their group algebras over fields of characteristic p. Understanding which properties of p-groups are determined by their group algebra is central to addressing this problem.
Which group-theoretic property of a finite p-group is guaranteed to be determined by its group algebra over the prime field ${{\mathbb {F}}}_p$?
1) The order of every subgroup
2) The nilpotency class of the group
3) The number of elements of each order
4) The exponent of the group
5) The number of normal subgroups
6) The isomorphism type of the maximal abelian direct factor
7) The number of non-abelian simple factors
✓ Correct Answer:
The correct answer is 6) The isomorphism type of the maximal abelian direct factor.
📚 Reference Text:
Title: The Modular Isomorphism Problem and Abelian Direct Factors Year: 2022 Paper ID: e9004fdbe0b1701981a0fe9c1591d77047e477bd Source: semantic-scholar URL: https://www.semanticscholar.org/paper/e9004fdbe0b1701981a0fe9c1591d77047e477bd Abstract: Let p be a prime and let G be a finite p -group. We show that the isomorphism type of the maximal abelian direct factor of G , as well as the isomorphism type of the group algebra over $${{\mathbb {F}}}_p$$ F p of the non-abelian remaining direct factor, if existing, are determined by $${{\mathbb {F}}}_p G$$ F p G , generalizing the main result in Margolis et al. (Abelian invariants and a reduction theorem for the modular isomorphism problem, Journal of Algebra 636 , 533-559 (2023)) over the prime field. To do this, we address the problem of finding characteristic subgroups of G such that their relative augmentation ideals depend only on the k -algebra structure of kG , where k is any field of characteristic p , and relate it to the modular isomorphism problem, extending and reproving some known results.
Question 33multiple-choice
Quantum algorithms have revolutionized the way certain mathematical problems are solved, particularly in the domain of group-based cryptography. The Hidden Subgroup Problem (HSP) plays a pivotal role in determining the security of cryptosystems against quantum attacks.
Which statement most accurately reflects the current understanding of efficient quantum algorithms for solving the Hidden Subgroup Problem (HSP) in various group structures?
1) Efficient quantum algorithms exist for HSP in all infinite groups commonly used in cryptography.
2) Efficient quantum algorithms for HSP are known for non-abelian groups but not for abelian groups.
3) Efficient quantum algorithms for HSP exist for finite abelian groups, but not for infinite or non-abelian groups.
4) Efficient quantum algorithms for HSP exist for braid groups and other infinite groups.
5) Efficient quantum algorithms for HSP are universally applicable to all group-based cryptosystems.
6) Efficient quantum algorithms for HSP are limited to finite non-abelian groups used in cryptography.
7) Efficient quantum algorithms for HSP do not exist for any group structure considered in cryptography.
✓ Correct Answer:
The correct answer is 3) Efficient quantum algorithms for HSP exist for finite abelian groups, but not for infinite or non-abelian groups..
📚 Reference Text:
Title: The Hidden Subgroup Problem and Post-quantum Group-based Cryptography Year: 2018 Paper ID: 1e52cc8602fcd382d760418cc4227a6df059e1b4 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/1e52cc8602fcd382d760418cc4227a6df059e1b4 Abstract: In this paper we discuss the Hidden Subgroup Problem (HSP) in relation to post-quantum group-based cryptography. We review the relationship between HSP and other computational problems discuss an optimal solution method, and review the known results about the quantum complexity of HSP. We also overview some platforms for group-based cryptosystems. Notably, efficient algorithms for solving HSP in such infinite group platforms are not yet known.
Question 34multiple-choice
In number theory, the multiplicative group modulo n, denoted (Z/nZ)×, encodes the structure of invertible elements under multiplication modulo n. The classification of its Sylow p-subgroups and the properties of its subgroups have important implications in algebra, cryptography, and analytic number theory.
Which of the following statements best describes a "maximally non-cyclic" multiplicative group (Z/nZ)×?
1) Every Sylow p-subgroup is cyclic for all p dividing n.
2) (Z/nZ)× is isomorphic to a cyclic group for all n.
3) All its subgroups are trivial.
4) All prime-power subgroups are elementary abelian groups, i.e., direct products of cyclic groups of order p.
5) Its order is always a prime number.
6) It contains no elements of order greater than 2.
7) It is isomorphic to the additive group of integers modulo n.
✓ Correct Answer:
The correct answer is 4) All prime-power subgroups are elementary abelian groups, i.e., direct products of cyclic groups of order p..
📚 Reference Text:
Title: Counting multiplicative groups with prescribed subgroups Year: 2020 Paper ID: 078a9baabae55abdd530d2d2afd48d4ce1714165 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/078a9baabae55abdd530d2d2afd48d4ce1714165 Abstract: We examine two counting problems that seem very group-theoretic on the surface but, on closer examination, turn out to concern integers with restrictions on their prime factors. First, given an odd prime [Formula: see text] and a finite abelian [Formula: see text]-group [Formula: see text], we consider the set of integers [Formula: see text] such that the Sylow [Formula: see text]-subgroup of the multiplicative group [Formula: see text] is isomorphic to [Formula: see text]. We show that the counting function of this set of integers is asymptotic to [Formula: see text] for explicit constants [Formula: see text] and [Formula: see text] depending on [Formula: see text] and [Formula: see text]. Second, we consider the set of integers [Formula: see text] such that the multiplicative group [Formula: see text] is “maximally non-cyclic”, that is, such that all of its prime-power subgroups are elementary groups. We show that the counting function of this set of integers is asymptotic to [Formula: see text] for an explicit constant [Formula: see text], where [Formula: see text] is Artin’s constant. As it turns out, both of these group-theoretic problems can be reduced to problems of counting integers with restrictions on their prime factors, allowing them to be addressed by classical techniques of analytic number theory.
Question 35multiple-choice
Quantum algorithms for hidden shift problems often exploit group structures to achieve improved efficiency, particularly for groups relevant to cryptography and computational number theory. Efficient space and time complexity are crucial for practical implementation on quantum hardware.
For the hidden shift problem over groups of the form $\mathbb{Z}_{2^t}^n$, which statement accurately describes the space complexity of a quantum algorithm that achieves polynomial running time in the dimension $n$?
1) Both classical and quantum space are exponential in $n$
2) Classical space is quadratic in $n$, while quantum space is linear in $n \log(k)$
3) Classical space is linear in $n$ and quantum space is quadratic in $n$
4) Both classical and quantum space are linear in $n$
5) Classical space is constant and quantum space is exponential in $n$
6) Classical space is cubic in $n$ and quantum space is linear in $n$
7) Classical space is linear in $n \log(k)$ and quantum space is quadratic in $n$
✓ Correct Answer:
The correct answer is 2) Classical space is quadratic in $n$, while quantum space is linear in $n \log(k)$.
📚 Reference Text:
Title: A new quantum algorithm for the hidden shift problem in $\mathbb{Z}_{2^t}^n$ Year: 2021 Paper ID: 7b395536481c005a536f85aa6df157ac978ab57c Source: semantic-scholar URL: https://www.semanticscholar.org/paper/7b395536481c005a536f85aa6df157ac978ab57c Abstract: In this paper we make a step towards a time and space efficient algorithm for the hidden shift problem for groups of the form $\mathbb{Z}_k^n$. We give a solution to the case when $k$ is a power of 2, which has polynomial running time in $n$, and only uses quadratic classical, and linear quantum space in $n\log (k)$. It can be a useful tool in the general case of the hidden shift and hidden subgroup problems too, since one of the main algorithms made to solve them can use this algorithm as a subroutine in its recursive steps, making it more efficient in some instances.
Question 36multiple-choice
In computational quantum chemistry, efficient electronic structure calculations often rely on representing Kohn-Sham orbitals using localized basis sets on real-space grids. Methods for constructing such bases leverage mathematical properties of the density matrix and linear algebra techniques to enhance computational scalability.
Which approach directly constructs localized orbitals by selecting linearly independent, localized columns from the density matrix using QR factorization with column pivoting, particularly optimizing for insulating systems?
1) Maximally Localized Wannier Functions (MLWFs)
2) Generalized Eigenvalue Decomposition (GED)
3) Density Functional Perturbation Theory (DFPT)
4) Projector-Augmented Wave (PAW) Method
5) Selected Columns of the Density Matrix (SCDM)
6) Fast Fourier Transform (FFT) Basis Construction
7) Linearized Augmented Plane Wave (LAPW) Method
✓ Correct Answer:
The correct answer is 5) Selected Columns of the Density Matrix (SCDM).
📚 Reference Text:
^ iis often localized around an atom or a chemical bond. Hence working with ^ i's can reduce both the storage and the computational cost. Assume we have access to ^ j(r)'s evaluated at a set of discrete grid points frigN i=1. Letf!igN i=1be a set of positive integration weights associated with the grid points frigN i=1, then the discrete orthonormality condition is given by (2.3)NX i=1^ j(ri)^ j0(ri)!i=jj0: Let^ j= [^ j(r1);^ j(r2);:::; ^ j(rN)]Tbe a column vector, and ^ = [ ^ 1;:::; ^ ne] be a matrix of size Nne. We call ^ the real space representation of the Kohn-Sham orbitals and de ne diagonal matrix W= diag[!1;:::;! N]. Our method requires the Kohn-Sham orbitals to be be represented on a set of real space grid points. This is the case for a plane-wave basis set, as well as other representations such as nite di erences, nite elements and wavelets. For instance, if the Kohn-Sham orbitals are represented using the plane-wave basis functions, their real space representation can be obtained on a uniform grid eciently with the fast Fourier transform (FFT) technique and in this case !itakes the same constant value for alli. It is in this setting that our method is of particular interest. However, 4 this procedure is also applicable to other basis sets such as Gaussian type orbitals or numerical atomic orbitals when a real space representation of the basis functions is readily available. Therefore, our method is amenable to most electronic structure software packages. We de ne = W1 2^ such that the discrete orthonormality condition in Eq. (2.3) becomes =I, whereIis an identity matrix of size ne. We now seek a compressed basis for the span of , denoted by the set of vectors = [ 1;:::; ne] where each i is a sparse vector with spatially localized support after truncating entries with small magnitudes. In such case, iis called a localized vector. 2.2. Selected columns of the density matrix. As opposed to widely-used procedures such as MLWFs [29], the key di erence in the SCDM procedure is that the localized orbitals iare obtained directly from columns of the density matrix P= . The aforementioned nearsightedness principle states that, for insulating systems, each column of the matrix Pis localized. As a result, selecting any linearly independent subset of neof them will yield a localized basis for the span of :However, pickingnerandom columns of Pmay result in a poorly conditioned basis if, for example, there is too much overlap between the selected columns. Therefore, we would like a means for choosing a well conditioned set of columns, denoted C= fc1;c2;:::;c neg;to use as the localized basis. Intuitively we expect such a basis to select columns to minimize overlaps with each other when possible. This is algorithmically accomplished with a QRCP factorization (see, e.g., [17]). More speci cally, given a matrix Aa QRCP seeks to compute a permutation matrix such that the leading sub-matrices ( A)1;:::;k; :for any applicable kare as
Question 37multiple-choice
In multipartite quantum information theory, the explicit construction and classification of locally maximally entangled (LME) states play a crucial role in understanding the structure of quantum state spaces. These constructions often rely on symmetry, arithmetic progressions, and tools like Schmidt decomposition and local unitary transformations.
For triples of the form (2, b, b) with b ≥ 2 in the classification of LME states, what is the dimension of the quotient space of local unitary orbits corresponding to these states?
1) 0
2) b
3) b+1
4) b-3
5) 2b-1
6) b-2
7) 1
✓ Correct Answer:
The correct answer is 4) b-3.
📚 Reference Text:
The conditions (5.13) imply b=c, so we find S2={(b,b)|b≥2}. The proposition then gives that the non-empty triples are (2 ,fi,fi+1) with (f0,f1) = (b,2b) or (f0,f1) = (b,b). Since the sequence is arithmetic, we have explicitly that (2 ,b,b) withb≥2 and (2,kb,(k+ 1)b) withbk>2. Proposition 5.3, implies that the quotient is a point except in the cas e (2,b,b), where it has dimension b−3. This reproduces the results that we obtained by our explicit constr uction in section 2. 38 Acknowledgements We would like to thank Jason Bell, Patrick Hayden, Alex May, Robert Ra ussendorf, David Stephen, and Michael Walter for helpful comments and discussions . This work is supported in part by the Natural Sciences and Engineering Research Council o f Canada and by the Simons Foundation. A Explicitconstructionofall LME statesfor (d1,d2,d3) = (2,B,C) In this appendix, we provide details of the explicit construction of all LME states with (d1,d2,d3) = (2,B,C) with 2≤B≤C. In this case, the necessary conditions (2.6) require that B≤C≤2B . (A.1) Using the Schmidt decomposition, and performing a U(2) rotation on the first factor, any state|Ψ/an}b∇acket∇i}htwith our desired properties can be written as |Ψ/an}b∇acket∇i}ht=1√ 2|1/an}b∇acket∇i}ht⊗ψ1 bc|b/an}b∇acket∇i}ht⊗|c/an}b∇acket∇i}ht+1√ 2|2/an}b∇acket∇i}ht⊗ψ2 bc|b/an}b∇acket∇i}ht⊗|c/an}b∇acket∇i}ht (A.2) whereψ1 bc|b/an}b∇acket∇i}ht⊗|c/an}b∇acket∇i}htandψ2 bc|b/an}b∇acket∇i}ht⊗|c/an}b∇acket∇i}htdefine orthonormal states of HB⊗HCand summation over bandcis implied. Making use of the Schmidt decomposition on ψ1 bc|b/an}b∇acket∇i}ht ⊗ |c/an}b∇acket∇i}htand performing U(B) andU(C) rotations on the second and third factors, we can write ψ1 bc=/bracketleftbig D{√pi}0B×(C−B)/bracketrightbig ψ2 bc=/bracketleftbig IB×BJB×(C−B)/bracketrightbig (A.3) whereD{√pi}is the diagonal matrix with elements√pi. Orthogonality of ψ1andψ2requires that tr(ID{√pi}) = 0. (A.4) The condition that ρBis maximally mixed gives 1 2D{pi}+1 2II†+1 2JJ†=1 B1 1B×B (A.5) while the condition that ρCis maximally mixed gives 1 2D{pi}+1 2I†I=1 C1 1B×B (A.6) I†J= 0 (A.7) 1 2J†J=1 C1 1(C−B)×(C−B). (A.8) 39 CaseC=B We begin with the special case C=B. Here, the conditions collapse to II†=I†I=D{2 B−pi} (A.9) together with the normalization condition (A.4). Defining Hermitian ma trices H+=1 2(I+I†)H−=−i 2(I −I†) (A.10) the first equality in (A.9) gives [ H+,H−] = 0, so the matrices are simultaneously diagonal- izable. We can write H1=UD1U†H2=UD2U†(A.11) so that I=UD{zi}U†(A.12) whereD{zi}is some general complex diagonal matrix. The latter equality in (A.9) g ives that UD{|zi|2}U†=D{2 B−pi}. (A.13) Without loss of generality, we can assume that the eigenvalues piare ordered from largest to smallest and the eigenvalues in D{|zi|2}are ordered from smallest to largest. Then we must have zi=eiφi/radicalbigg 2 B−pi (A.14) andUmust commute with D{pi}. It istherefore block-diagonal, with blocks corresponding to blocksofequaleigenvaluesin D{pi}. However, recallingthatthelocalunitarytransformations on the two subsystems of size Bact as D{pi}→W†D{pi}V I →W†IV (A.15) we see that whenever such blocks exist, we can take local unitary t ransformations with W=V=U, to eliminate U, leaving I=D{eiφi√ 2 B−pi}(A.16) Wheneverpi= 0, we have residual local unitary transformations that can be us ed to set φi= 0 also. We can also set φi= 0 whenpi=2 B. Finally, the condition (A.4) gives that /summationdisplay ieiφi/radicalbigg pi(2 B−pi) = 0 (A.17) so we must choose the phases so that the complex numbers in the su m add to
Question 38multiple-choice
Quantum computing algorithms leverage group theory to solve certain problems exponentially faster than classical algorithms, with the Hidden Subgroup Problem (HSP) occupying a central role in algorithmic breakthroughs and ongoing research. The distinction between abelian and nonabelian group structures is crucial for understanding the complexity and applicability of quantum solutions.
Which problem, formulated as a nonabelian Hidden Subgroup Problem, has significant implications for complexity theory and cryptography due to the current lack of efficient quantum algorithms?
1) Integer factorization
2) Discrete logarithm computation
3) Pell’s equation solution
4) Class group computation
5) Graph isomorphism
6) Hidden translation in abelian groups
7) Stabilizer problem
✓ Correct Answer:
The correct answer is 5) Graph isomorphism.
📚 Reference Text:
Sorbonne Paris-Cit´ e, Paris, France 75205 (frederic. magniez@univ-paris-diderot.fr). ¶C N R S ,L I A F A ,U n i v e r s i t ´ e Paris Diderot, Sorbonne Paris-Cit´ e, Paris, France 75205, and Cen- tre for Quantum Technologies, National University of Singapore, Singapore (miklos.santha@liafa.univ-paris-diderot.fr). /bardblSchool of Technology and Computer Science, Tata Institute of Fundamental Research, Mumbai, India 400005 (pgdsen@tcs.tifr.res.in). 1 2 FRIEDL, IVANYOS, MAGNIEZ, SANTHA, AND SEN While no classical algorithm can solve this problem with polynomial query com- plexityevenif Gisabelian, thebiggestsuccessofqua ntumcomputinguntilnowisthat it can be solved by a quantum algorithmefficiently for any abelian G. We will refer to this quantum algorithm as the standard algorithmfor Hidden Subgroup .T h em a i n tool for this solutionis Fourier sampling based on the (approximate)quantum Fourier transform for abelian groups which can be efficiently implemented quantumly [29]. Simon’s XOR-mask finding [42], Shor’s fac torization and discrete logarithm finding algorithms [41], and Kitaev’s algorithm [29] for the abelian stabilizer problem are all special cases of this general solution. Quantum algorithms of Hallgren [20, 21] andSchmidt and Vollmer [40] computing class groups and unit groups of number fields, including the solution of Pell’s equation, also follow along these lines. Finding an efficient algorithm for Hidden Subgroup for nonabelian groups G is considered to be one of the most important challenges at present in quantum com- puting. Besides its intrinsic mathematical interest, the importance of this problem is enhanced by the fact that it contains as a special case the graph isomorphism problem. Unfortunately, although its query complexity is shown to be polynomial by Ettinger, Høyer, and Knill [14], nonabelian Hidden Subgroup seems to be much more difficult than the abelian case. Although considerable effort was spent on it in the last few years, only a small number of successes can be reported. They can be di- vided into twocategories. The standardabelianFourier samplingbasedalgorithmhasbeen extended to some nonabelian groups in [39, 22, 19, 16, 33, 12] using the quantum Fouriertransformoverthese (nonabelian) groups. Although efficient quantum Fourier transform implementations are known for several nonabelian groups [8, 23, 37, 32], the power of the technique appears to be very limited. In a different approach, Hid- den Subgroup was efficiently solved in the context of specific nonabelian black-box groups [5, 45] by [26] without using the Fourier transform on the group, and instead using Fourier transforms over abelian groups only. Similarly, only abelian Fourier transforms were used by [24, 6, 10, 27, 28] to solve the hidden subgroup problem insome specific kinds of nonabelian groups. See [11] for a more detailed review of hidden subgroup algorithms and related problems. In light of the apparent hardness of Hidden Subgroup in nonabelian groups, a natural line of research is to address subproblems of Hidden Subgroup which, in some groups, capture the main difficulty of the original problem. In a pioneeringpaper, Ettinger and Høyer [13], in the case of dihedral groups, implicitly considered another paradigmatic group problem, Hidden Translation .H e r ew ea r eg i
Question 39multiple-choice
Quantum algorithms for lattice problems often rely on Fourier analysis over lattices and their quotient spaces, revealing structural information in the frequency domain. The interpretation of quantum measurement outcomes depends critically on probability distributions derived from normalized quantum states.
In the quantum Fourier transform of a function defined over a lattice, what ensures that the weights assigned to dual lattice points correspond to valid probability values in quantum measurement?
1) The function is normalized to unit norm so that the squared magnitudes of Fourier coefficients are nonnegative and sum to one.
2) The dual lattice is chosen to be orthogonal to the original lattice basis vectors.
3) The Fourier transform is computed only over even-dimensional lattices.
4) The quantum state is post-selected on the origin of the dual lattice.
5) The normalization is performed after measurement to ensure proper probabilities.
6) The original function is required to be real-valued for valid probabilistic interpretation.
7) The computation uses only integer-valued lattice points.
✓ Correct Answer:
The correct answer is 1) The function is normalized to unit norm so that the squared magnitudes of Fourier coefficients are nonnegative and sum to one..
📚 Reference Text:
Fourier transform of ψδ andψ, respectively: pδ(y) =˙bψδ(y)˛˛bψδ(y)¸ withbψδ=FδZmψ; p(y) =˙bψ(y)˛˛bψ(y)¸ withbψ=FRnψ. (6.8) It can be shown that pδis close top. Let us now focus on the distribution p(y) =˙bψ(y)˛˛bψ(y)¸ . We have ψ=wf,bψ=bw∗(FRmf), wherebw=FRmw. Sincefis a hidden subgroup oracle, we may regard it as a function on Rm/Land definebf=FRm/Lf. That is, bfu=Z Rm/Le2πi/angbracketleftx,u/angbracketrightf(x)dx foru∈L∗, f(x) =1 d(L)X u∈L∗e−2πi/angbracketleftx,u/angbracketrightbfu. (6.9) 301 The Fourier transform over Rmis (FRmf)(y) =Z Rme2πi/angbracketleftx,y/angbracketright 1 d(L)X u∈L∗e−2πi/angbracketleftx,u/angbracketrightbfu! dx =1 d(L)X u∈L∗bfuδ(y−u). It follows that bψ(y) =` bw∗(FRmf)´ (y) =1 d(L)X u∈L∗bfubw(y−u),(6.10) p(y) =1 d(L)2X u,u/prime∈L∗˙bfu/prime˛˛bfu¸ bw(y−u)bw(y−u/prime).(6.11) The last equation is complicated, but to have a good ap- proximation it is enough to keep the terms with u=u/prime. Let us consider the quantum state FRm|ψ/angbracketrightwhose wavefunction is given by Eq. (6.10). It consists of identically shaped peaks at the points u∈L∗. Each peak has a weight qu=˙bfu˛˛bfu¸ d(L)2=1 d(L)2Z e2πi/angbracketleftx−x/prime,u/angbracketright˙ f(x/prime)˛˛f(x)¸ dxdx/prime, where the integral is over (Rm/L)2. The numbers qucan be interpreted as probabilities because they are nonnegative and add up to 1. Indeed, let us normalize fto make a function of unit norm, g(x) =d(L)−1/2f(x). Then X u∈L∗qu=1 d(L)X u∈L∗˙ bgu˛˛bgu¸ =˙ bg˛˛bg¸ =/angbracketleftg|g/angbracketright= 1. 7. REFERENCES [Ban93] W. Banaszczyk. New bounds in some transference theorems in the geometry of numbers. Mathematische Annalen, 296(1):625–635, 1993. [BK93] Johannes Buchmann and Volker Kessler. Computing a reduced lattice basis from a generating system, 1993. Preprint, August 4, 1993. [BP87] Johannes Buchmann and Michael Pohst. Computing a lattice basis from a system of generating vectors. In Eurocal’87 , volume 378 ofLNCS, pages 54–63. Springer-Verlag, June 1987. [BV11] Zvika Brakerski and Vinod Vaikuntanathan. Fully homomorphic encryption from ring-LWE and security for key dependent messages. In Advances in cryptology—CRYPTO 2011 , volume 6841 of LNCS, pages 505–524. Springer, 2011. [Coh93] Henri Cohen. A course in computational algebraic number theory. Springer-Verlag New York, Inc., New York, NY, USA, 1993. [FIM+03] Katalin Friedl, Gabor Ivanyos, Frederic Magniez, Miklos Santha, and Pranab Sen. Hidden translation and orbit coset in quantum computing. In Proceedings of the Thirty-Fifth Annual ACM Symposium on Theory of Computing, San Diego, CA, 9–11June 2003. [GH11] C. Gentry and S. Halevi. Implementing gentry’s fully-homomorphic encryption scheme. Eurocrypt 2011 , pages 132–150, 2011.[GKP01] Daniel Gottesman, Alexei Kitaev, and John Preskill. Encoding a qubit in an oscillator. Phys. Rev. A , 64:012310, Jun 2001. [GVL96] Gene H. Golub and Charles F. Van Loan. Matrix Computations . Johns Hopkins University Press, Baltimore, MD, 3rd edition, 1996. [Hal05] Sean Hallgren. Fast quantum algorithms for computing the unit group and class group of a number field. In Proceedings of the 37th Annual ACM Symposium on Theory of Computing, pages 468–474, 2005. [Hal07] Sean Hallgren. Polynomial-time quantum algorithms for Pell’s equation and the principal ideal problem. Journal of the ACM, 54(1):1–19, 2007. [HMR+10] Sean Hallgren, Cristopher Moore, Martin R¨otteler, Alexander Russell, and Pranab Sen. Limitations of quantum coset states for graph isomorphism. J. ACM, 57:34:1–34:33, November 2010. [Kit95] Alexei Kitaev. Quantum measurements and the abelian stabilizer problem, 1995. quant-ph/9511026. [KW08] Alexei Kitaev and William A. Webb.
Question 40multiple-choice
In the representation theory of complex reductive algebraic groups such as GL(n), highest weight theory provides a systematic approach to classifying irreducible representations. The moment map and related techniques play a foundational role in geometric invariant theory and have important computational and theoretical applications.
Which of the following statements best characterizes the role of highest weight vectors in the decomposition of representations of GL(n) and their connection to irreducibility?
1) Highest weight vectors are invariant under all diagonal matrices but not under upper-triangular matrices.
2) Highest weight vectors correspond to the lowest eigenvalues of the torus action in any representation.
3) Highest weight vectors are orthogonal to all other weight vectors in the weight space decomposition.
4) Highest weight vectors exist only for representations of abelian groups, not reductive groups.
5) Highest weight vectors determine the trace of representation matrices but not their irreducibility.
6) Highest weight vectors are not preserved under dualization of representations.
7) Highest weight vectors are eigenvectors under the action of the Borel subgroup, and their weights uniquely characterize irreducible representations.
✓ Correct Answer:
The correct answer is 7) Highest weight vectors are eigenvectors under the action of the Borel subgroup, and their weights uniquely characterize irreducible representations..
📚 Reference Text:
supported by NSF grant CCF-1412958. 16 2 Geometric invariant theory In this section, we present some results from geometric invariant theory that will feature centrally in the analysis of our algorithm in Section 3. While stated for tensors, all results in this section can easilybeextendedtoarbitraryrationalrepresentationsofconnectedcomplexreductivealgebraic groups. Most of the results are well known and only some are new. All previously known results will be citedwith references and we will make suretohighlight the new components. Section 2.1 discussesbasicsofthehighestweighttheory. Section2.2givesaformaldefinitionofthemoment mapandalsodiscussesthesocalled“shiftingtrick"thatreducestheproblemofmembershipin moment polytopes to a null cone problem. Section 2.3 considers degree bounds for highest weight vectorswhichareusedtoboundtheinitialrandomnessusedinAlgorithm1. Section2.4recallsa classical construction of highest weight vectors and uses this construction to prove bounds on their evaluations (crucial in the analysis of Algorithm 1). Section 2.5 develops a necessary and sufficient condition for Borel scalability (i.e., scaling using tuples of upper-triangular matrices). Asbefore,let GGLpn1q GLpndq,KUpn1qUpndq,VTenpn0;n1;:::;ndq Cn0bCn1b:::bCnd, and XPpVqaG-stable irreducible projective subvariety (e.g., an orbit closure). 2.1 Highest weight theory We first recall the representation theory of GLpnq(see, e.g., [ FH13] for an introduction). Let W be a finite-dimensional GLpnq-representation, equipped with a Upnq-invariant inner product. Let TpnqGLpnqdenotethesubgroupconsistingofinvertiblediagonalmatrices,calledthe maximal torusofGLpnq. SinceTpnqis commutative, its action can be jointly diagonalized. Thus, any finite-dimensional GLpnq-representation Wcanbewrittenasadirectsumofso-called weightspaces , WÀ !Wp!q, whereTpnqacts on any vector wPWp!qasTw !pTqwfor allTPTpnq. Here,!is an integer vector and !pTq ±n j1T!j j;j. We write pWqfor the set of all weights that occur in W. Now letBpnqGLpnqdenote the Borel subgroup of invertible upper-triangular matrices, which contains Tpnq. Ahighest weight vector is a vectorwPWthat is an eigenvector of the Bpnq-action. Letdenote its weight, which is now called highest weight . Necessarily, 1¥¥n, i.e.,is ordered non-increasingly, and we have that Rw pRqwfor allRPBpnq, where pRq ±n j1Rj j;j. We denote by HWVpWqthe space of highest weight vectors in Wwith highest weight . Theirreducible representations of GLpnqcontain a unique (up to scalar multiple) highest weightvector andare characterized byits highestweight. Wewrite Vfor theirreducible representation (which we always equip with a K-invariant inner product, denoted x;y) andv forahighestweightvector(whichwechoosetobeofunitnorm). Thus, HWVpVqCvif, andzerootherwise. Itisknownthat Bt0xv;exppAtqvytrrAdiagpqsforallnn-matricesA. It can also be verified that GLpnqrvsUpnqrvs(in particular, this G-orbit is closed). The dual ofanirreduciblerepresentationisalsoirreduciblewithhighestweight Ò,sothatV V. We now consider the group GGLpn1q GLpndq. All the preceding notions generalize immediately by considering tuples or tensor products of the relevant objects, and we shall use similar notation. Thus, the maximal torus isTTpn1qTpndq, theBorel subgroup is 17 BBpn1qBpndq.Highest weight vectors satisfy Rw pRqw;where pRqd¹ i1ni¹ j1pRpiq j;jqpiq j (7) for all tuples R pRp1q;:::;Rpdqq PB, andweight vectors satisfy the same relation restricted toTB.Weightsandhighest weight are now tuples pp1q;:::;pdqqof integer vectors as before. Thesums°ni j1piq jarenecessarilyequalfor i1;:::;d,andwewilldenotethemby jj. Thus,{jjPP pn1;:::;ndq. We denote by HWVpWqthe space of highest weight vectors in aG-representation W. The irreducible representations of Gare again labeled by their highest weight and denoted by V. Indeed, they are simply given by tensor products of the corresponding GLpniq-representations,i.e., VVp1qb:::bVpdq;thesameholdsfortheirhighestweightvectors. For every tuple of matrices ApAp1q;:::;Apdqq(Apiqisnini), we have that Bt0xv;exppAtqvyn¸ i1trrApiqdiagppiqqs; (8) where exppAtq:exppAp1qtqb:::bexppApdqtq. As before, wewrite ppp1qq;:::;pp1qqq, so thatV V. 2.2 Moment map and shifting trick LetWbe aG-representation. The associated
Question 41multiple-choice
Molecular simulation force fields are essential tools in computational chemistry for predicting physical properties such as bulk density and vaporization enthalpy. The accuracy of these models depends significantly on which types of atomic interactions they include.
Which of the following best explains why omitting three-body interaction terms in a force field leads to systematic overestimation of bulk density and underestimation of vaporization enthalpy in organic molecules?
1) It causes the modeled molecules to adopt higher-energy conformations, reducing intermolecular attractions.
2) It leads to excessive hydrogen bonding, inflating both density and enthalpy values.
3) It neglects long-range electrostatic interactions, making the system less cohesive.
4) It fails to account for collective atomic interactions that moderate total system energy, resulting in denser packing and reduced energy required for vaporization.
5) It incorrectly assigns partial charges, creating artificial dipole moments.
6) It omits van der Waals forces entirely, causing molecules to repel each other.
7) It introduces quantum tunneling effects that disrupt normal phase behavior.
✓ Correct Answer:
The correct answer is 4) It fails to account for collective atomic interactions that moderate total system energy, resulting in denser packing and reduced energy required for vaporization..
📚 Reference Text:
the a priori origin of the former and the fact the the experimental bulk de nsity is one of the target properties on which transferable FFs are tu ned. Molecule ρ(kg/m3) GAFF17OPLS17CGenFF59QMD-FF Exp. C1 1375 1373 - 1420 147980- 148083 C2 1258 1200 1241 1328 131680- 132779 C3 1054 1176 1217 1261 121379 C4(273 K) 898 915 912 969 92480 N1 -1410 - 1439 144079 N2 -1265 - 1378 136979 B1 1326 1740 - 1813 166283- 167579 B2 1946 2474 2402 2570 246979- 248480 B3 1807 2335 2170 2311 216983- 217979 B4 1229 1408 1398 1446 134580- 135479 I1 -2042 - 2107 193679 I2 1599 1835 - 1818 169583- 173780 St. Dev. (kg/m3)254 94 47 98 - Table 7: Bulk density ( ρ, kg/m3): comparison between experiment and data computed from MD simulations performed with popular FFs and the QMD-FF. Unle ss otherwise stated, all values refer to T = 298 K. The other property usually employed in empirical FF tuning i s the vaporization en- thalpy. In Table 8 the ∆ Hvapcomputed and experimental values are compared for each compound. By looking at the standard deviations, reported a s usual in the last row, it 27 Molecule ∆Hvap(kJ/mol) GAFF17OPLS17CGenFF59QMD-FF Exp. C1 28.4 29.2 - 30.4 31.384 C2 26.5 23.4 25.3 31.028.879-29.985 C3 24.9 24.3 29.9 27.7 26.579 C4(273 K) 22.9 26.9 24.0 28.9 27.886 N1 -35.8 - 42.3 41.770 N2 -35.5 - 34.5 33.070 B1 18.5 25.2 - 25.5 22.879 B2 28.6 34.2 43.1 40.3 37.484 B3 34.7 48.8 40.1 51.3 41.784 B4 31.4 34.8 35.3 39.4 32.279 I1 -29.0 - 39.8 32.079 I2 33.7 33.8 - 40.6 34.179 St. Dev. (kJ/mol) 4.6 3.7 3.7 4.8 - Table 8: Vaporization enthalpy (∆ Hvap, kJ/mol): comparison between experiment and data computed from MD simulations performed with popular FFs and the QMD-F F. Unless otherwise stated, all values refer to T = 298 K. appears as the QMD-FF error is slightly larger than the one co mputed with GAFF and ∼1 kJ/mol worse than OPLS and CGenFF. Consistently with ρ, where most values were overestimated with respect of the experiment, there is a gen eral overestimation of the total energy of the system given by the QMD-FF. In the previou s parameterizations per- formed on pyridine32and benzene40molecules, this issue was traced back to the absence of three body effects, given the pure two-body nature of the QM D-FF. As a matter of fact, the effect of the inclusion of three-body interactions ,intheFFshasbeenrecently reportedtoin the FF, which has been recently44quantified to account for 15-20% of the total interaction energy of water bulk phase (hence causing not negligible inaccuracies), was found to contribute only for a 2-5%, when organic molecul es, more similar to the ones here considered, are taken into account.36In turn, consistently with the present results, tha absence of three-body effects was reported36to cause a ∼5% overestimation of the bulk density and to lower the ∆ Hvapby 6 to 12%. Here, itthese inacurracies could be also connected to a only partial
Question 42multiple-choice
Quantum computing architectures must efficiently implement universal gate sets while addressing error correction and hardware compatibility. Parity encoding offers unique advantages for logical operations and physical qubit connectivity.
Which characteristic of parity encoding directly enables virtual all-to-all connectivity among logical qubits in quantum computing architectures?
1) Encoding logical qubit information in the parity of multiple physical qubits
2) Restricting interactions to nearest-neighbor qubits only
3) Utilizing only non-diagonal multi-qubit operators
4) Requiring enforcement of parity constraints at every computational step
5) Implementing logical gates exclusively with multi-qubit entangling operations
6) Limiting error correction resources to single-qubit bit-flip detection
7) Designing the architecture for quantum annealing without adaptation for universal computation
✓ Correct Answer:
The correct answer is 1) Encoding logical qubit information in the parity of multiple physical qubits.
📚 Reference Text:
Universal Parity Quantum Computing Michael Fellner ,1,2,*Anette Messinger ,2Kilian Ender ,1,2and Wolfgang Lechner1,2,† 1Institute for Theoretical Physics, University of Innsbruck, A-6020 Innsbruck, Austria 2Parity Quantum Computing GmbH, A-6020 Innsbruck, Austria (Received 19 May 2022; accepted 16 September 2022; published 27 October 2022) We propose a universal gate set for quantum computing with all-to-all connectivity and intrinsic robustness to bit-flip errors based on parity encoding. We show that logical controlled phase gate and Rz rotations can be implemented in parity encoding with single-qubit operations. Together with logical Rx rotations, implemented via nearest-neighbor controlled-NOT gates and an Rxrotation, these form a universal gate set. As the controlled phase gate requires only single-qubit rotations, the proposed schemehas advantages for several cornerstone quantum algorithms, e.g., the quantum Fourier transform. We present a method to switch between different encoding variants via partial on-the-fly encoding and decoding. DOI: 10.1103/PhysRevLett.129.180503 Designing quantum computers [1–17] and quantum algorithms [18–26]is a current grand challenge in science and engineering, motivated by the prospect of solving certain problems exponentially faster than any known classical algorithms [21]. However, the fundamental rules of quantum mechanics that make this new paradigm possible also impose fundamental restrictions. In contrast to classical information, quantum information cannot becopied, which is known as the no-cloning theorem, but only propagated [27]. Thus, quantum computers will not be able to follow the von Neumann architecture [28]with separated memory and computational unit. As the quantum CPUserves as memory and computational unit at the same time, connectivity between any quantum bits on the chip is required. In current standard approaches to gate-basedquantum computers, either these long-range interactions are implemented as physical interactions, which limits scalability, or quantum information is moved on the chipvia SWAP sequences, which requires a large overhead in gates. Although there are recent approaches toward qubit routing that address this issue [29,30] , exchanging infor- mation between qubits remains a challenging problem. In this Letter, we propose a novel universal quantum computing approach based on the Lechner-Hauke-Zoller (LHZ) architecture [31], which was originally designed for quantum annealing. In this parity-based paradigm, eachphysical qubit represents the parity of multiple logicalqubits. We extend the LHZ architecture, up to now onlyused for solving combinatorial optimization problems, to a universal quantum computing approach by providing auniversal gate set on parity-encoded states and with that, open up new possibilities for universal quantum compu- tation. These extensions include an additional row of data qubits added to the original LHZ layout to enable control of single logical qubits. We introduce logical operations, inparticular R xrotations, to establish a universal gate set in the logical space. As the parity constraints no longer needto be enforced throughout the computation, they can beutilized for error correction. As it only requires nearest-neighbor interactions between qubits on a square lattice chip, our proposal can beimplemented on state-of-the-art quantum devices, indepen-dent of the qubit platform. Suitable platforms are forexample superconducting qubits [15,32 –36], neutral atoms [37–39], or trapped ions [40–43]. We show that the parity transformation renders diagonal multiqubit operatorsbetween arbitrary logical
Question 43multiple-choice
Shifted quantum affine algebras and their representations are central objects in modern mathematical physics, with deep connections to quantum integrable systems, cluster algebras, and supersymmetric gauge theories. These structures often involve studying categories of modules and their interplay with algebraic geometry and dualities.
Which mathematical structure is conjectured to parametrize simple modules for non simply-laced truncations of shifted quantum affine algebras, supported by evidence from Baxter polynomiality in quantum integrable models?
1) The Weyl group of the original Lie algebra
2) The Langlands dual Lie algebra
3) The quantum torus algebra
4) The Hecke algebra of type A
5) The root lattice of the original Lie algebra
6) The classical universal enveloping algebra
7) The category of perverse sheaves on the flag variety
✓ Correct Answer:
The correct answer is 2) The Langlands dual Lie algebra.
📚 Reference Text:
Title: Representations of shifted quantum affine algebras Year: 2020 Paper ID: cb1fb8aa6c65366a57400881d34c9299c6007a11 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/cb1fb8aa6c65366a57400881d34c9299c6007a11 Abstract: We develop the representation theory of shifted quantum affine algebras $\mathcal{U}_q^\mu(\hat{\mathfrak{g}})$ and of their truncations which appeared in the study of quantized K-theoretic Coulomb branches of 3d $N = 4$ SUSY quiver gauge theories. Our direct approach is based on relations that we establish with the category $\mathcal{O}$ of representations of the quantum affine Borel algebra $\mathcal{U}_q(\hat{\mathfrak{b}})$ and on associated quantum integrable models we have previously studied. We introduce the category $\mathcal{O}^\mu$ of representations of $\mathcal{U}_q^\mu(\hat{\mathfrak{g}})$ and we classify its simple objects. For $\mathfrak{g} = sl_2$ we prove the existence of evaluation morphisms to $q$-oscillator algebras. We establish the existence of a fusion product and we get a ring structure on the sum of the Grothendieck groups $K_0(\mathcal{O}^\mu)$. We introduce induction and restriction functors to the category $\mathcal{O}$ of $\mathcal{U}_q(\mathfrak{b})$. As a by product we classify simple finite-dimensional representations of $\mathcal{U}_q^\mu(\hat{\mathfrak{g}})$ and we obtain a cluster algebra structure on the Grothendieck ring of finite-dimensional representations. We establish a necessary condition for a simple representation to descend to a truncation, which is also sufficient for $\mathfrak{g} = sl_2$. We introduce a related partial ordering on simple modules and we prove a truncation has only a finite number of simple representations. We state a conjecture on the parametrization of simple modules of a non simply-laced truncation in terms of the Langlands dual Lie algebra. We have several evidences, including a general result for simple finite-dimensional representations proved by using the Baxter polynomiality of quantum integrable models.
Question 44multiple-choice
In representation theory and quantum computation, the symmetric group Sn captures the symmetries of n indistinguishable objects, with its representations acting on spaces like (C^2)⊗n for qubits. Irreducible representations of Sn are classified by integer partitions and have important combinatorial and physical implications.
Which of the following statements correctly describes the classification of irreducible representations of the symmetric group Sn?
1) Each irreducible representation corresponds to a unique element of Sn written in cycle notation.
2) Irreducible representations are classified by the set of all possible two-row Young diagrams for n.
3) Every irreducible representation is associated with a distinct Pauli operator on n qubits.
4) The number of irreducible representations equals 2^n, the dimension of the n-qubit Hilbert space.
5) Irreducible representations are determined solely by the number of cycles in a permutation.
6) Irreducible representations of Sn are classified by the integer partitions of n, often depicted using Young diagrams.
7) Each irreducible representation corresponds directly to a swap operator permuting two qubits.
✓ Correct Answer:
The correct answer is 6) Irreducible representations of Sn are classified by the integer partitions of n, often depicted using Young diagrams..
📚 Reference Text:
Representations of the Symmetric Group In this section we review standard facts about representations of the symmetric group which will be necessary to prove our results. 2.2.1 Preliminary definitions For any finite-dimensional vector space V, we use GL(V)to denote the group of invertible linear transformations from Vto itself. We use Snto denote the symmetric group, the group of permutations of nobjects. We will specify elements group using cycle notation, withedenoting the identity element. Recall that a representation of Snis a group homo- morphismρ:Sn→GL(V). The vector space Vis also referred to as an Sn-module or simply a module. The group algebra C[Sn]can be defined as an Sn-module as follows. Promote the elements{π1,π2,...,πn!}∈Snto basis vectors π1,π2,...,π n!with the multiplication rule πiπj=πkifπiπj=πk. Then C[Sn]is given by C[Sn] ={c1π1+c2π2+···+cn!πn!|cj∈C} (2.6) whereci∈Cfor alli, andSnacts on C[Sn]by left-multiplication. A representation (ρ,V)ofSnalso gives rise to a representation of the algebra C[Sn]via the homomorphism ˜ρ:C[Sn]→C[GL(V)]defined by its action ˜ρ/parenleftiggn!/summationdisplay i=1ciπi/parenrightigg =n!/summationdisplay i=1ciρ(πi). (2.7) It is common to use ρto refer to the representation of both the group and its group algebra. ASn-module in general decomposes into a number of Sn-submodules . A module V isirreducible if the only submodules of Vare itself and the trivial module {0}. For a Accepted in Quantum2024-03-25, click title to verify. Published under CC-BY 4.0. 10 representation (ρ,V)of any finite group, it follows from Maschke’s Theorem that there exists a decomposition of Vas V=/circleplusdisplay λVλ (2.8) where each Vλis an irreducible module. We will be interested in matrix representations ρofSnwhereVis(C2)⊗n, which is to say thatρ(π)is a2n×2nmatrix for each π∈Sn. Maschke’s Theorem then implies that there is an invertible matrix Usuch that ρ(π) =U/bracketleftigg/circleplusdisplay λρλ(π)/bracketrightigg U−1, π∈Sn, (2.9) whereρλare irreducible representations or irreps of Sn. The decomposition of Eq. (2.8) into irreducible modules can thus be seen as a block-diagonalization of matrices in ρ(C[Sn])⊆ M2n(C). 2.2.2 Irreducible representations of Sn Irreducible representations of Snare in one-to-one correspondence with integer partitions ofn, λ= [λ1,λ2,...,λk], λ 1≥λ2≥···≥λk>0,k/summationdisplay i=1λi=n (2.10) which we will denote as λ⊢n. It is convenient to associate the partition λwith the Young diagram of shapeλ, which consists of krows indexed top to bottom such that the i-th row containsλiboxes. As an example, the partition [3,2]corresponds to the shape (2.11) Of crucial importance to us later in the paper will be those irreps ρ[n−k,k]that correspond to two row Young diagrams and the respective modules V[n−k,k], defined for k= 0,...,⌊n 2⌋. 2.3 Swap Matrices and Permutations In order to analyze the QMC Hamiltonian, we introduce the swap matrices Swapijas permutations on n-qubit states. Definition 2.2. The swap matrices Swapijare defined by their action on tensor products of|ψ1⟩,|ψ2⟩,...,|ψn⟩∈C2as follows: Swapij/parenleftig |ψ1⟩⊗···⊗|ψi⟩⊗···⊗|ψj⟩⊗···⊗|ψn⟩/parenrightig =|ψ1⟩⊗···⊗|ψj⟩⊗···⊗|ψi⟩⊗···⊗|ψn⟩. (2.12) The connection to Quantum Max Cut is made by noticing that the Swapijis the following linear combination of the Pauli matrices Swapij=1 2/parenleftig I+σi Xσj X+σi Yσj Y+σi Zσj Z/parenrightig . (2.13) Accepted in Quantum2024-03-25, click title to verify. Published under CC-BY 4.0. 11 Proposition 2.3. The QMC Hamiltonian HGof Eq. (2.5) in terms of the Swapij’s defined in Eq. (2.12) is given
Question 45multiple-choice
Quantum processors can be designed using multi-level quantum systems called qudits, which offer increased computational capacity compared to traditional qubits. Silicon-photonic integrated circuits enable scalable and robust implementation of such quantum devices.
Which of the following features most directly enables a programmable quantum processor based on ququarts to achieve enhanced computational parallelism and improved resource efficiency compared to qubit-based systems?
1) Employing error-correcting codes optimized for binary logic gates
2) Utilizing superconducting transmon qubits for state manipulation
3) Integrating nitrogen-vacancy centers in diamond for coherence
4) Implementing single-photon detectors for readout
5) Encoding quantum information in four-level qudits, increasing information density per particle
6) Incorporating trapped ions with microwave control pulses
7) Using classical post-processing to simulate quantum algorithms
✓ Correct Answer:
The correct answer is 5) Encoding quantum information in four-level qudits, increasing information density per particle.
📚 Reference Text:
Title: A programmable qudit-based quantum processor Year: 2022 Paper ID: a4390a7f1c46ff8a7fdae863295d0b610023d977 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/a4390a7f1c46ff8a7fdae863295d0b610023d977 Abstract: Controlling and programming quantum devices to process quantum information by the unit of quantum dit, i.e., qudit, provides the possibilities for noise-resilient quantum communications, delicate quantum molecular simulations, and efficient quantum computations, showing great potential to enhance the capabilities of qubit-based quantum technologies. Here, we report a programmable qudit-based quantum processor in silicon-photonic integrated circuits and demonstrate its enhancement of quantum computational parallelism. The processor monolithically integrates all the key functionalities and capabilities of initialisation, manipulation, and measurement of the two quantum quart (ququart) states and multi-value quantum-controlled logic gates with high-level fidelities. By reprogramming the configuration of the processor, we implemented the most basic quantum Fourier transform algorithms, all in quaternary, to benchmark the enhancement of quantum parallelism using qudits, which include generalised Deutsch-Jozsa and Bernstein-Vazirani algorithms, quaternary phase estimation and fast factorization algorithms. The monolithic integration and high programmability have allowed the implementations of more than one million high-fidelity preparations, operations and projections of qudit states in the processor. Our work shows an integrated photonic quantum technology for qudit-based quantum computing with enhanced capacity, accuracy, and efficiency, which could lead to the acceleration of building a large-scale quantum computer. Qudit-based quantum devices can outperform qubit-based ones, but a programmable qudit-based quantum computing device is still missing. Here, the authors fill this gap using a programmable silicon photonic chip employing ququart-based encoding, showing the scaling advantages compared to the qubit counterpart.
Question 46multiple-choice
In quantum information theory, the concept of unitary t-designs is crucial for simulating random quantum operations and understanding the universality of gate sets. Representation theory of unitary groups underpins the mathematical structure of these designs and their connection to universal quantum computation.
Which condition guarantees that an infinite closed subgroup G of U(d) is universal for quantum computation via unitary gates?
1) G contains all diagonal unitary matrices.
2) G is abelian and acts transitively on V.
3) G is a unitary 1-design and includes the identity.
4) G is a unitary 2-design and contains SU(d).
5) G consists only of anti-symmetric tensors.
6) G preserves the symmetric decomposition of V⊗t for all t.
7) G commutes with all elements in U(d).
✓ Correct Answer:
The correct answer is 4) G is a unitary 2-design and contains SU(d)..
📚 Reference Text:
in Proposition 3. LetVbe ad-dimensional vector space, and let U(V)be the corresponding unitary group, the group of all unitary trans- formations of V. By picking a basis, we may identify VwithCdandU(V)withU(d). As discussed above, we have the natural representation (τ,V)ofU(V)onV, where τis simply the identity map. This representation τis in fact irreducible. We then take the tensor product of tcopies of the natural representation, determining the tensor product representation (or diagonal representation )(τt,V⊗t), given by τt(U) =U⊗t. This has a decomposition into irreps for U(V), V⊗t∼=M iV⊕ni i, (B1) where, as above, each Viis irreducible and Vi∼=Vjiffi=j. Proposition 3 is concerned with the sum of the multiplicities,P ini. For t= 1,Vis already irreducible, as mentioned above, soP in1= 1. For t= 2,V⊗2decomposes into a direct sum of two inequivalent irreps, namely the symmetric and anti-symmetric tensors, soP ini= 2. Now let Gbe a closed subgroup G⊆U(V). The representation V⊗trestricts to a representation of G, which then has its own decomposition into irreps for G. In fact, it suffices to consider the restriction of each Vito a representation of G. The 23 content of Proposition 3 is that Gis at-design if and only if, for each Viappearing with nonzero multiplicity in (B1), Viis also irreducible as a representation of G. 3. Proof of Theorem 7 We now restate and prove Theorem 7. Theorem 10. LetV∼=Cd, with d≥2. LetGbe an infinite closed subgroup of U(d) =U(V). IfGis a unitary 2-design, thenSU(d)⊆G. In particular, any set of unitaries generating Gis a universal set. Proof. Since Gis an infinite closed subgroup of U(d), the closed subgroup theorem implies that it is an infinite compact Lie group, with a nonzero complexified Lie algebra g. By compactness, the Peter-Weyl theorem implies that G’s finite- dimensional representations are completely reducible. Since we assume that Gis a2-design, it is also a 1-design; which implies Vis an irrep for G(recall Proposition 3). Since dimV=d >1, this implies that Gis not abelian: equivalently, theG-subrepresentation [g,g]is nonzero. Further, since commutators are traceless, we have 0̸= [g,g]⊆sl(d). These are the main facts we will need below. By Proposition 3, since Gis a2-design, V ⊗ V decomposes into a direct sum of exactly two irreps for G. Note that by Schur’s lemma, this is equivalent to the following condition on the commutant: dimC(EndG(V ⊗ V )) = 2 .We now apply a chain of standard vector space isomorphisms: EndG(V ⊗ V )∼=((V ⊗ V )⊗(V ⊗ V )∗)G∼=((V ⊗ V∗)⊗(V ⊗ V∗)∗)G∼=EndG(V ⊗ V∗). (B2) In particular, this implies dimCEndG(V ⊗ V∗) = 2 .Now we follow Ref. 7 to show that this condition gives universality. Applying Schur’s lemma in the same way as above, if V ⊗ V∗∼=L iV⊕mi i is a decomposition of V ⊗ V∗into distinct irreps for G, then 2 = dim CEndG(V ⊗ V∗) =X im2 i. Since the miare positive integers, we conclude that V ⊗V∗decomposes into exactly two irreducible representations for G, each of multiplicity 1. Further, we can identify them exactly. Note that V ⊗
Question 47multiple-choice
In quantum information processing, error characterization and correction often involve advanced mathematical techniques to ensure the physical validity of quantum operations. One important approach is fitting experimental quantum processes to models that satisfy complete positivity and trace-preservation constraints.
Which of the following statements best explains the effect of imposing CPTP (completely positive, trace-preserving) constraints on an experimentally determined quantum supermatrix?
1) It eliminates all eigenvalues, making the supermatrix singular.
2) It causes the largest Kraus operator to vanish, altering the dominant process behavior.
3) It increases the number of negative eigenvalues, indicating reduced physical validity.
4) It ensures the operation is physically realizable and typically improves correlation with theoretical and simulated models.
5) It randomizes the action of the quantum superoperator, reducing benchmarking accuracy.
6) It removes all rotation-based errors without affecting correlation.
7) It transforms the supermatrix into a classical probability matrix without quantum properties.
✓ Correct Answer:
The correct answer is 4) It ensures the operation is physically realizable and typically improves correlation with theoretical and simulated models..
📚 Reference Text:
U1,expandU1,simincreased from 0.90 to 0.96 on left-multiplying U1by the Hermitian conjugate of this product of single-spin rotation operators, these rotations are the cumulative result of many small rotation errors and are not simply traced back to any single short-coming in the expe riments. For com- pleteness, the axes and angles of the single-spin rotations that be st fit the error operator U∆=U† 1,simU1,expare also shown in Table III, together with the eigenvalues of the cor re- sponding corrected superoperators in Fig. 16. The product of th ese rotations similarly has a 0.97 correlation with U∆, but in this case one must right-multiply U1,expby the Hermitian conjugate of the product to correct it. The close co-incidence be tween the angles of rotation about the x-axis on spin 1 in the first case and about the z-axis on spin 3 in the latter case is expected, since UQFTσ1 xU† QFT=σ3 z(recallσ1 xσ3 zis a fixed point of the QFT). Finally, it is of interest to demonstrate that despite a substantial n umber of negative eigenvalues in the Choi matrix of the experimental supermatrix, it is not necessary to change TABLE III: The two triples of single-spin rotations that optimize the c orrelation coefficient between the best unitary approximations to the largest Krau s operators of the experimental and simulated supermatrices, when the experi mental supermatrix is left or right multiplied by the unitary matrix corresponding to e ach one of these triples. Side Spin x,y&z-Directional Cosines of Rotation Axis Rotation Angle left 1 −0.992 0 .007 0 .123 36.7◦ left 2 −0.386 −0.243 0 .890 9.0◦ left 3 0.703 0 .701 −0.123 16.9◦ right 1 0.059 −0.031 0 .998 14.2◦ right 2 0.227 −0.512 0 .829 10.2◦ right 3 −0.092 −0.292 −0.952 38.3◦ 24 it much in order to obtain a supermatrix which represents a complete ly positive and trace- preserving superoperator. For this reason the supermatrix MCPTPwhich best-fit the Choi matrix of the experimental supermatrix subject to the constrain t that it was both positive semidefinite and satisfied the trace-preservation conditions was c omputed as described in Section III. Although this procedure made essentially no change in t he largest Kraus op- erator (as expected), it did have a significant effect on the experim ental supermatrix as a whole. The correlations between this CPTP-fit and the other super matrices that we have dealt with up to now are given in Table IV, along with those to the origina l experimental supermatrix for comparison. This shows that even though imposing the complete positivity constraint on the experimental observations did not change the s upermatrix very much, the change was distinctly in the right direction since it improved the corre lation with both the simulated and theoretical supermatrices. This is further confirme d by Fig. 17, which shows the eigenvalues of MCPTP, along with those of the superoperator ¯U1,CPTP⊗U1,CPTPobtained from the best unitary approximation to its largest Kraus operator (scaled down so as to have the same trace as MCPTP), and those of
Question 48multiple-choice
In quantum mechanics and linear algebra, the anti-symmetric subspace of a tensor product space is crucial for describing the behavior of fermions and constructing multi-particle wavefunctions. The anti-symmetric projector operator is central to extracting states that obey the Pauli exclusion principle.
Which of the following statements about the anti-symmetric subspace ASymₖ(ℂᵈ) and its orthogonal projector P(d, k)_asym is correct?
1) The dimension of ASymₖ(ℂᵈ) is always dᵏ, regardless of k and d.
2) The anti-symmetric projector operator is neither idempotent nor self-adjoint.
3) Basis vectors of the anti-symmetric subspace can include repeated indices.
4) For d < k, the dimension of ASymₖ(ℂᵈ) equals k! (k factorial).
5) The anti-symmetric subspace is used primarily to describe bosons.
6) P(d, k)_asym projects onto the symmetric subspace.
7) The dimension of ASymₖ(ℂᵈ) is zero if d < k, and equals the binomial coefficient "d choose k" otherwise.
✓ Correct Answer:
The correct answer is 7) The dimension of ASymₖ(ℂᵈ) is zero if d < k, and equals the binomial coefficient "d choose k" otherwise..
📚 Reference Text:
introduce the anti-symmetric subspace. Definition 18 (Anti-symmetric subspace) .The anti-symmetric subspace is the set: ASymk(Cd):=/braceleftig |ψ⟩∈(Cd)⊗k:Vd(π)|ψ⟩= sgn (π)|ψ⟩ ∀π∈Sk/bracerightig , (74) where sgn (σ)denotes the sign of a permutation σ∈Sk. Similarly as before, we can define the operator: P(d,k) asym :=1 k!/summationdisplay π∈Sksgn (π)Vd(π). (75) and prove the following theorem: Theorem 19. The operator P(d,k) asym is the orthogonal projector on the anti-symmetric subspace ASymk(Cd). Proof. We have: Vd(σ)P(d,k) asym =1 k!/summationdisplay π∈Sksgn (π)Vd(σ)Vd(π) =1 k!/summationdisplay π∈Sksgn (π)Vd(σπ) (76) =1 k!/summationdisplay π∈Sksgn/parenleftbig σ−1π/parenrightbig Vd(π) = sgn/parenleftbig σ−1/parenrightbig P(d,k) asym (77) = sgn (σ)P(d,k) asym. (78) Similarly,P(d,k) asymVd(σ) = sgn (σ)P(d,k) asym. Using this, we can show that P(d,k)2 asym =P(d,k) asym: P2 asym =1 k!/summationdisplay π∈Sksgn (π)Vd(π)P(d,k) asym =1 k!/summationdisplay π∈Sk(sgn (π))2P(d,k) asym =P(d,k) asym. (79) We also have P(d,k)† asym =P(d,k) asym: P(d,k)† asym =1 k!/summationdisplay π∈Sksgn (π)V† d(π) =1 k!/summationdisplay π∈Sksgn (π)Vd(π−1) =1 k!/summationdisplay π∈Sksgn/parenleftbig π−1/parenrightbig Vd(π−1) (80) =1 k!/summationdisplay π∈Sksgn (π)Vd(π) =P(d,k) asym. (81) We can show that Im/parenleftig P(d,k) asym/parenrightig ⊆ASymk(Cd) as follows: for all |ψ⟩∈(Cd)⊗k, we haveP(d,k) asym|ψ⟩∈ ASymk(Cd), sinceVd(π)P(d,k) asym|ψ⟩= sgn (π)P(d,k) asym|ψ⟩for allπ∈Sk, where we used again that Vd(π)P(d,k) asym = sgn (π)P(d,k) asym. Moreover, we can also show that ASymk(Cd)⊆Im/parenleftig P(d,k) asym/parenrightig . In fact, if|ψ⟩∈ASymk(Cd), then we have: P(d,k) asym|ψ⟩=1 k!/summationdisplay π∈Sksgn (π)Vd(π)|ψ⟩=1 k!/summationdisplay π∈Sk(sgn (π))2|ψ⟩=|ψ⟩. (82) Therefore, we have shown that Im/parenleftig P(d,k) asym/parenrightig = ASymk(Cd), which implies that P(d,k) asym is the orthogonal projector on the anti-symmetric subspace ASymk(Cd). Next, we compute the dimension of the anti-symmetric subspace. Proposition 20 (Dimension of the anti-symmetric subspace) .Ifd≥k, we have: Tr/parenleftig P(d,k) asym/parenrightig = dim/parenleftbig ASymk(Cd)/parenrightbig =/parenleftbiggd k/parenrightbigg , (83) otherwise Tr/parenleftig P(d,k) asym/parenrightig = 0. 11 Proof. First, we have: ASymk(Cd) = Im/parenleftig P(d,k) asym/parenrightig = span/parenleftig P(d,k) asym|i1⟩⊗···⊗|ik⟩:∀i1,...,ik∈[d]/parenrightig . (84) To count the number of linearly independent vectors P(d,k) asym|i1⟩⊗···⊗|ik⟩, we observe that if there are at least two tensor factors of |i1⟩⊗···⊗|ik⟩with matching entry i.e. there exist l̸=m∈[k] such that il=im, then: P(d,k) asym|i1⟩⊗···⊗|ik⟩= 0. (85) This because: P(d,k) asym|i1⟩⊗···⊗|ik⟩=P(d,k) asymVd(τl,m)|i1⟩⊗···⊗|ik⟩ (86) = sgn (τl,m)P(d,k) asym|i1⟩⊗···⊗|ik⟩ (87) =−P(d,k) asym|i1⟩⊗···⊗|ik⟩, (88) where in the second equality we used P(d,k) asymVd(τl,m) = sgn (τl,m)P(d,k) asym (as shown in the proof of Theorem 19). Therefore, i1,...,ikmust all be distinct for P(d,k) asym|i1⟩⊗···⊗|ik⟩to be nonzero. This also implies that if d < k , then Tr/parenleftig P(d,k) asym/parenrightig = 0. Therefore, we now focus on the case d≥k. If for all m∈[d], there exist two vectors |i1⟩⊗···⊗|ik⟩and|j1⟩⊗···⊗|jk⟩with i1,j1...,ik,jk∈[d] such that both sets, {i1,...,ik}and{j1,...,jk}, containnm∈{0,1}elements equal tom, then there exists a permutation Vd(π) such that Vd(π)|i1⟩⊗···⊗|ik⟩=|j1⟩⊗···⊗|jk⟩. SinceP(d,k) asymVd(π) = sgn (π)P(d,k) asym, we haveP(d,k) asym|i1⟩⊗···⊗|ik⟩= sgn (π)P(d,k) asym|j1⟩⊗···⊗|jk⟩, and hence, P(d,k) asym|i1⟩⊗···⊗|ik⟩andP(d,k) asym|j1⟩⊗···⊗|jk⟩are linearly dependent. Thus, taking n1,...,nd∈{0,1}withn1+···+nd=k, we define|n1,...,nd⟩asP(d,k) asym|i1⟩⊗ ···⊗|ik⟩, where|i1⟩⊗···⊗|ik⟩is any computational basis vector such that, for each m∈[d], there existnmelementsj∈[k] such that ij=m. It is easy to see that such vectors are orthogonal and hence independent. Therefore, the dimension is given by the number of such independent vectors, which is equal to the number of ways to choose an
Question 49multiple-choice
Quantum speed limits define the minimum time required to perform specific quantum gates, which is crucial for optimizing operations in neutral atom quantum computing. Various control configurations, such as sequential, parallel, and phase, influence both the achievable gate speed and error rates when implementing gates like the controlled-Z (CZ) gate.
Which control configuration for the CZ gate in neutral atom systems both achieves the quantum speed limit of 350 ns and requires the fewest physical resources while automatically mapping the |↓↓⟩ state onto itself?
1) Sequential configuration with site-dependent amplitude controls
2) Parallel configuration with simultaneous amplitude variation
3) Sequential configuration with phase and amplitude modulation
4) Phase configuration with intrinsic state mapping
5) Parallel configuration with independent phase controls
6) Sequential configuration with non-site-specific fields
7) Parallel configuration with optimized initial guess fields
✓ Correct Answer:
The correct answer is 4) Phase configuration with intrinsic state mapping.
📚 Reference Text:
ZZZ( γ) and ZZZZ( γ), cf. Eq. ( A6), have been obtained for the maximally entangling gate at γ=π/4. The results for the CNOT gate on neutral atoms have been obtained using site-dependent control fields for each atom. Eq. ( A6) and Ref. [ 65]. Its QSL TCZ QSL=700 ns is twice as long compared to the other two configurations. In order to compare the results from the three configura- tions in terms of how successful the optimization has been infinding solutions, the histogram on the right side of Fig. 3(a) provides the probability density for obtaining final errors ε T within certain ranges. For obtaining a solution with errors εT/lessorequalslantεmax, we find the lowest probability for the “sequen- tial” configuration (diamonds)—coinciding with the highestQSL—and the highest and almost identical probability forthe other two configurations. Among those two, the “phase”configuration (crosses) needs to be emphasized in particular.From a physical perspective, setting /Omega1 ↓(t) andϕ↓(t) to zero automatically ensures that |↓↓/angbracketright is mapped onto itself—as required by the CZ gate. This is not automatically guaranteedby the other two configurations and the optimization needsto explicitly ensure it and therefore needs to solve a slightlymore complex optimization problem. However, the advan-tage of having one basis state automatically mapped correctlydoes not translate into an advantage regarding the QSL ofT CZ QSL=350 ns or the reachable error in general. Interestingly, both the “parallel” and “phase” configurations yield the sameachievable lowest errors for each gate time T. This is visually highlighted by the lines connecting the lowest errors per T in Fig. 3(a). Moreover, it should be stressed that these errors εTare reached for almost every set of initial guess fields, i.e., independent of the initial starting point within the problem’scontrol landscape, and obtained independently for both con-figurations. This supports the conjecture that T CZ QSL=350 ns is the actual QSL for a CZ gate and generally validates ourmethod in determining the QSL. From an optimal controlperspective, it is interesting to see that the flexibility origi-nating from the extended set of available control fields in the“parallel” configuration can not be turned into an advantagein error or time compared to the “phase” configuration. Froma practical perspective, the latter is advantageous for experi-mental realizations as it requires fewer physical resources. While the three configurations discussed so far should only be viewed as examples, we did not find any configuration giv-ing rise to faster CZ gates. Hence, we assume T CZ QSL=350 ns to be the fundamental QSL across all configurations. A naturalcomparison for T CZ QSLwith a value from the literature would be the gate time from the analytical protocol introduced inRef. [ 20], especially because it uses the same control fields as 023026-9 BASILEWITSCH, DLASKA, AND LECHNER PHYSICAL REVIEW RESEARCH 6, 023026 (2024) the “phase” configuration to implement the gate. For our pa- rameters, we find TCZ lit≈340 ns as analytical gate time, which we assume identical with our QSL TCZ QSL=350 ns given the rather coarse sampling of gate times Tin Fig. 3(a). However, it should
Question 50multiple-choice
Quantum sensing utilizes advanced protocols to enhance measurement sensitivity and spectral resolution, particularly when detecting complex signals composed of multiple frequencies. The choice of phase estimation algorithm significantly impacts both dynamic range and experimental efficiency.
Which approach enables unambiguous phase measurement with high sensitivity in a single readout, leveraging quantum entanglement to demultiplex multiple spin signals onto distinct qubit outputs?
1) Classical Fourier transform analysis on a single qubit system
2) Standard Ramsey interferometry with repeated measurements
3) Machine learning-based iterative phase estimation
4) Bayesian adaptive phase estimation with multiple interrogations
5) Direct amplitude sampling using analog signal processing
6) Inverse quantum Fourier transform protocol implemented on a quantum register
7) Frequency filtering via classical lock-in amplification
✓ Correct Answer:
The correct answer is 6) Inverse quantum Fourier transform protocol implemented on a quantum register.
📚 Reference Text:
measured. To achieve the best possible sensitivity in each of thetwo phase acquisition steps interrogation times τclose to the decoherence limit T 2of the sensor are used, which results inacquiring a maximum phase ϕ. On the other hand, such choice of τcould result in ϕmore than πand thereby in an ambiguity in determining the actual strength of the signal. This becomes ultimately important when dealing with a high dynamic range correlation signal with multiple frequency components of variousstrengths. In the case of a nanoscale NMR signal, which iscomposed of multiple frequencies f iand amplitudes aithis results in a beating of the free precession signal requiring a high dynamic range sensing method (see Fig. 1b). Thus, the problem at hand is to design a phase correlation measurement protocol that at the same time yields a large dynamic range and allows for high spectral resolution (i.e., long correlation times Tc). This known dynamic range - sensitivity1trade-off could be tackled by a family of phase estimation algorithms, including Bayessian, adaptive and machine learning sensing protocols1–8and the ones based on inverse quantum Fourier transform (QFT)9–11. As correla- tion measurements typically have a large time overhead ( Tc= 10–2 0m s )i nc o m p a r i s o nt ot h es e n s i n gt i m e s( τ=10–100μs)12(also see SI for times used in present manuscript), repetitive measure-ments within each correlation step required in the adaptive techniques may not be well suitable. Further, including the in fluence of the quantum back-action in such repetitive measurements leadsto erroneous results on the measured dynamics 13,14.O nt h eo t h e r hand, phase estimation using the inverse QFT could resolve ambiguities in each run (measurement) of the protocol and provide a sensitivity that is only limited by the coherence time of the sensor.In this work, we advance the correlation measurement protocol with phase estimation based on the inverse QFT algorithm using the NV center and a three nuclear spin register. We utilize the QFT †and QFT as a convenient transformation to convert the acquired phase to the population basis of the register. To demonstrate and benchmark the scope of QFT-based phase estimation we perform here threedifferent experiments using both classical and quantum fields. We first start by demonstrating the phase to population mapping by QFT and show how it differs from the standard Ramsey measure- ments which do the same. After this we benchmark the QFT † algorithm and perform high dynamic range sensing of an arti ficial classical AC-signal generated by the RF source. Later we extend it to 13. Physikalisches Institut, IQST and Centre for Applied Quantum Technologies, University of Stuttgart, Stuttgart, Germany.2Max-Planck Institute for Solid State Research, Stuttgart, Germany.✉email: v.vorobyov@pi3.uni-stuttgart.de; j.wrachtrup@pi3.uni-stuttgart.dewww.nature.com/npjqi Published in partnership with The University of New South Wales1234567890():,; the correlation spectroscopy protocol of the nuclear spin-bath and demonstrate the demultiplexing of target spin-signals onto separate register qubits outputs. As the sensor readout performed only at the end of the protocol, an entanglement
Question 51multiple-choice
Quantum algorithms have dramatically impacted the security landscape of cryptography, especially for schemes relying on algebraic problems such as the discrete logarithm. The distinction between groups and semigroups, and the computational complexity of related problems, is crucial for understanding potential quantum vulnerabilities.
In the context of quantum computing and algebraic structures, which statement best explains why some cryptosystems based on discrete logarithms in semigroups are not secure against quantum attacks, while certain generalizations of the discrete logarithm problem remain hard in semigroups?
1) Quantum algorithms cannot efficiently solve any discrete logarithm problems in semigroups.
2) Semigroups always have the same computational properties as groups regarding discrete logarithm problems.
3) The lack of inverses in semigroups makes all discrete logarithm problems easy for quantum computers.
4) Quantum algorithms like Shor's can efficiently solve standard discrete logarithm problems in semigroups, undermining cryptosystems, but certain generalized or shifted versions may still be hard due to the structure of semigroups.
5) The security of semigroup-based cryptosystems is guaranteed because quantum algorithms only accelerate group-based problems.
6) Discrete logarithm problems in semigroups are always harder than those in groups for quantum computers.
7) Semigroups do not admit any cryptographically hard problems in the presence of quantum algorithms.
✓ Correct Answer:
The correct answer is 4) Quantum algorithms like Shor's can efficiently solve standard discrete logarithm problems in semigroups, undermining cryptosystems, but certain generalized or shifted versions may still be hard due to the structure of semigroups..
📚 Reference Text:
Title: Quantum computation of discrete logarithms in semigroups Year: 2013 Paper ID: 3ed363357765b3362fdd98ae3fa589e5c4626cbc Source: semantic-scholar URL: https://www.semanticscholar.org/paper/3ed363357765b3362fdd98ae3fa589e5c4626cbc Abstract: Abstract We describe an efficient quantum algorithm for computing discrete logarithms in semigroups using Shor's algorithms for period finding and the discrete logarithm problem as subroutines. Thus proposed cryptosystems based on the presumed hardness of discrete logarithms in semigroups are insecure against quantum attacks. In contrast, we show that some generalizations of the discrete logarithm problem are hard in semigroups despite being easy in groups. We relate a shifted version of the discrete logarithm problem in semigroups to the dihedral hidden subgroup problem, and we show that the constructive membership problem with respect to k ≥ 2 generators in a black-box abelian semigroup of order N requires Θ ˜(N 1 2-1 2k )$\tilde{\Theta }(N^{\frac{1}{2}-\frac{1}{2k}})$ quantum queries.
Question 52multiple-choice
Computational spectroscopy often relies on advanced quantum chemical methods to predict vibrational frequencies and simulate molecular spectra, balancing accuracy with computational cost. Composite approaches can leverage the strengths of different methods for efficient and precise results, especially in the study of hydrogen-bonded systems and solvatochromic effects.
Which computational strategy most effectively reduces computational expense while maintaining accurate peak position predictions for vibrational spectra by combining a density functional theory approach for spectral shifts with a high-level coupled cluster method for reference frequencies?
1) Using only the M06-2X/cc-pVTZ density functional theory method for all calculations
2) Relying exclusively on the CCSD-F12b coupled cluster method for both spectral shifts and reference frequencies
3) Applying the QVP1 frozen-bath approximation without further refinement
4) Calculating spectral shifts and reference frequencies with the same low-level DFT method
5) Employing only ab initio molecular dynamics with CCSD throughout the simulation
6) Combining spectral shifts from CCSD-F12b and reference frequencies from M06-2X
7) Utilizing the composite CCSD/M06 method by applying DFT for spectral shifts and CCSD for reference frequencies
✓ Correct Answer:
The correct answer is 7) Utilizing the composite CCSD/M06 method by applying DFT for spectral shifts and CCSD for reference frequencies.
📚 Reference Text:
based on the QVP1 transition frequencies in green. The spectrum determined from the DAF shows a series of side peaks with an interval of about 50 cm−1, attributed to the coupling with rotations of the water molecule about the H–Cl axis. This effect is not included in the instantaneous frozen-bath approximation in the QVP1 model. Xue et al. Page 7 J Chem Theory Comput . Author manuscript; available in PMC 2018 February 01. Author Manuscript Author Manuscript Author Manuscript Author Manuscript The maximum peak position from the QVP1 method is located at 2653 cm−1, redshifted by 185 cm−1 from the strongest peak position (2838 cm−1) displayed on the spectrum based on the classical DAF (Table 1). Since M06-2X/cc-pVTZ was used in both approaches, consistent with the dynamic trajectory, the difference between the two methods is primarily due to nuclear quantum effects of the oscillator, which clearly have significant contributions to the computed vibrational frequencies in Figure 3a. Nevertheless, the full widths at half- maximum (fwhm) of the spectrum peak are similar from both methods (21 and 23 cm−1, respectively). Figure 3b compares the results obtained using three different electronic structure methods: M06-2X (green line), CCSD(T)-F12b (red line), and a composite of M06-2X spectral shift and the CCSD(T)-F12b frequency. In the latter approach, the CCSD(T)-F12b frequency for the reference state is used, but the spectral shift due to dynamic fluctuations of the solvent is determined at the M06-2X level using QVP1. We designate this method as CCSD(T)/M06 in Figure 3b, which significantly reduces the computational costs compared with that when the perturbation theory is also carried out at the CCSD(T)-F12b level. Figure 3b shows that the maximum peak position from QVP1 using CCSD(T)-F12b is only 14 cm−1 greater than the experimental value (2723.5 cm−1).16,17 The M06-2X frequency in the HCl(H 2O) complex is blue-shifted by 84.6 cm−1 from the coupled-cluster value, and this is mainly due to the difference in the reference state between the two methods: 80.6 cm−1 from 2650.5 cm−1 (CCSD(T)-F12b) to 2569.9 cm−1 (M06-2X). Thus, the solvatochromic shifts are not very sensitive to the quantum chemical models used, here, differing only by 4 cm−1 (84.6 vs 80.6 cm−1). Clearly, it can be a major time-saving approach to use density functional theory to estimate the spectral shift, but the reference frequency value is evaluated using coupled cluster theory (eq 1) since the latter only needs to be computed once. As the computational cost of the QVP method is equivalent to that required to determine the perturbation energy at the discrete points, allowing the instantaneous vibrational frequency of the oscillator to be determined on-the-fly of dynamics simulations. Indeed, the composite approach, employing the CCSD(T) zeroth order frequency (reference state) and QVP1 frequency shift from M06-2X (blue curve in Figure 3b), yields a maximum peak position at 2725 cm−1, only 13 cm−1 different from the CCSD(T)-F12b value (2738 cm−1). Interestingly, the composite result is in better accord with experimental results than CCSD(T)-F12b, perhaps due to fortuitous error cancellations. Furthermore, the computed
Question 53multiple-choice
In the classification of finite p-groups of nilpotency class 3, characteristic matrices over finite fields are used to distinguish group types and compute invariants related to subgroup structure. Arithmetic properties like quadratic residues can influence the existence of specific group extensions.
Which condition ensures that two finite p-groups of class 3 with characteristic matrices w and w(Ḡ) are isomorphic, based on matrix equivalence over the field Fp?
1) There exists a permutation matrix over Fp such that w(Ḡ) = P w P⁻¹
2) There exists an invertible diagonal matrix X over Fp such that w(Ḡ) = X w diag(x₁₁⁻¹ x₂₂⁻¹, x₁₁⁻² x₂₂⁻¹)
3) The trace of w and w(Ḡ) are equal modulo p
4) The determinant of w is a quadratic residue modulo p
5) A scalar multiple relates w(Ḡ) and w over Fp
6) The entries of w and w(Ḡ) are pairwise congruent modulo p
7) Both groups have identical commutator subgroup orders
✓ Correct Answer:
The correct answer is 2) There exists an invertible diagonal matrix X over Fp such that w(Ḡ) = X w diag(x₁₁⁻¹ x₂₂⁻¹, x₁₁⁻² x₂₂⁻¹).
📚 Reference Text:
types give non-isomorphic groups, (i i)G∼=¯Gif and only if there exists X= diag(x11,x22), an invertible matrix over Fp, such that w(¯G) = Xw(G)diag(x−1 11x−1 22,x−2 11x−1 22). By Table 5.2, we get the groups of Type (M1)–(M9). /square w(G) Case X w (¯G) Group Remark (a) w22/negationslash= 0diag(z,w11) where w 22=νz2/parenleftbigg 1w12w−1 11z−1 0 ν/parenrightbigg (M1) s=−w12w−1 11z−1 (a) w22= 0,w12/negationslash= 0 diag( w−1 11w12,w11)/parenleftbigg 1 1 0 0/parenrightbigg (M2) (a) w22=w12= 0 diag(1 ,w11)/parenleftbigg 1 0 0 0/parenrightbigg (M3) (b) w21/negationslash= 0 diag( w21,w−1 21w12)/parenleftbigg0 1 1 0/parenrightbigg (M4) (b) w21= 0 diag(1 ,w12)/parenleftbigg 0 1 0 0/parenrightbigg (M5) (c) w21/negationslash= 0 diag( w21,1)/parenleftbigg0 0 1w22w−2 21/parenrightbigg (M6) if w22/negationslash= 0 (M7) if w22= 0t=w−1 22w2 21 (c) w21= 0,w22/negationslash= 0diag(1,z) where w 22=νz2/parenleftbigg 0 0 0ν/parenrightbigg (M8) (c) w21=w22= 0/parenleftbigg 0 0 0 0/parenrightbigg (M9) Table 5.2: The types in Theorem 5.8 6 The case Φ(G′)≤G3∼=C2 p Suppose that Gis a finite p-group with Φ( G′)≤G3∼=C2 p,G3≤Z(G) andG/G3∼= Mp(n,m,1), where n >1 forp= 2 and n≥m. Let G/G3=/an}bracketle{t¯a,¯b,¯c|¯apn=¯bpm= ¯cp= 1,[¯c,¯a] = [¯b,¯c] = 1/an}bracketri}ht. 24 Then, without loss of generality, we may assume that G=/an}bracketle{ta,b,c/an}bracketri}htwhere [a,b] =c. Let x= [b,c],y= [c,a]. Then G3=/an}bracketle{tx,y/an}bracketri}ht. Sinceapn∈G3, we may assume that apn= xw11yw12. By similar reasons, we may assume that bpm=xw21yw22andcp=xw31yw32. Letw(G) =/parenleftbiggw11w12 w21w22/parenrightbigg andv(G) =/parenleftbiggw31 w32/parenrightbigg . Then we get two matrices over Fp. w(G) is called a characteristic matrix of G, andv(G) is called a characteristic vector ofG. Notice that w(G) andv(G) will be changed if we change the generators a,b. We also call a,ba set of characteristic generators of w(G) andv(G). Theorem 6.1. Suppose that Gis a finite p-group such that Φ(G′)≤G3∼=C2 p,G3≤ Z(G)andG/G3∼=Mp(n,m,1). Letw(G) = (wij)be a characteristic matrix of G. Then the following conclusions hold: (1)IfImin= 1, thenm= 1; (2)IfImax= 2, thenn≤2; (3)Ifp= 2andm= 1, thenImin= 1; (4)Ifp >3andn=m= 1, thenImin/ne}ationslash= 1if and only if w11=w22=w12+w21= 0; (5)Ifp >2andn > m= 1, thenImin/ne}ationslash= 1if and only if w11=w12= 0; (6)Ifp >3andn=m= 1, thenImax/ne}ationslash= 2if and only if (w12+w21)2−4w11w22is a quadratic non-residue modular p; (7)Ifp >2,m= 1andn= 2, thenImax= 2if and only if w22/ne}ationslash= 0; (8)Ifp >2,n=m= 2andΦ(G′) = 1, thenImax= 2if and only if (w12+w21)2− 4w11w22is a quadratic non-residue modular p; (9)Ifp >2,n=m= 2and(w31,w32) = (0,1), thenImax= 2if and only if w11/ne}ationslash= 0and one of the following holds: (i) (w12+w21)2−4w11(w22+1)is a quadratic non-residue modular p;(ii)w21=w11w22+w11and(w12+w21)2= 4w21. ProofLetN=/an}bracketle{tb,ap,c,x,y/an}bracketri}htandMi=/an}bracketle{tabi,bp,c,x,y/an}bracketri}htwhere 0≤i≤p−1. TheN andMiare all maximal subgroups of G. (1) Assume that His anA1-subgroup of index p. Then d(H/H′) = 2. Since G/G3∈ A1,H/G3is abelian and hence H′≤G3. It follows that d(H/G3) = 2. Since Φ(G)≤H,d(Φ(G)/G3)≤2. Since Φ( G)/G3=/an}bracketle{t¯ap,¯bp,¯c/an}bracketri}htis of type ( pn−1,pm−1,p), we havem= 1. (2) LetK=/an}bracketle{tb,c/an}bracketri}ht. ThenK∈ A1and|G:K| ≥ |G:KG3|=pn. SinceImax= 2, n≤2. (3) Ifm= 1 and p= 2, then 1 = [ a,b2] =c2x. Hence c2=x. In this case, M1=/an}bracketle{tab,c/an}bracketri}ht ∈ A1. ThusImin= 1. (4) Ifn=m= 1 and p >3, then|G|=p5and 1 = [ a,bp] =cp. IfImin/ne}ationslash= 1, thenN∈ A2andMi∈ A2. Sincey/ne}ationslash∈ /an}bracketle{tb,c/an}bracketri}ht,w22= 0. By calculation, [ abi,c] =y−1xi and (abi)p=apbip=xw11+iw21yw12. Hence, ∀i,w11+iw21=−iw12. It follows that w11=w12+w21=
Question 54multiple-choice
Multi-value (qudit-based) quantum computing utilizes quantum systems with more than two levels, enabling higher-dimensional encoding and potentially improved resource efficiency for quantum algorithms. Key algorithms like Deutsch-Jozsa and Bernstein-Vazirani can be generalized for these systems, offering new capabilities for function analysis and phase estimation.
In a quaternary (d=4) quantum system implementing generalized phase estimation, which advantage is achieved over binary (qubit) systems when measuring the eigenphase of a randomised gate?
1) The ability to achieve higher gate fidelity without error correction
2) Reduced decoherence due to shorter gate times
3) Elimination of the need for photon coincidence counting
4) Increased computational speed by removing the oracle query
5) Measurement in a single algorithmic interaction regardless of system dimension
6) Automatic correction of Poissonian statistical uncertainties
7) Achieving the same computational accuracy with fewer algorithmic interactions due to higher-dimensional encoding
✓ Correct Answer:
The correct answer is 7) Achieving the same computational accuracy with fewer algorithmic interactions due to higher-dimensional encoding.
📚 Reference Text:
The algorithm iteratively computes all mdits of the eigenphase backwardly, in which, notably, each dit is once estimated with d-ary accuracy. Figure 5b–d report measured eigenphases of 4-dimensional unitary matrices by quaternary phase estimation. We estimated the four eigenphases for three logic gates, i.e., a phase gate Z4, a Fourier gate F4Fig. 4 Implementations of generalised Deutsch-Jozsa and Bernstein- Vazirani algorithms in quaternary. a Quantum logical circuit for implementing the d-ary Deutsch-Jozsa and Bernstein-Vazirani algorithms. This circuit can be implemented by the scheme in Fig. 1a, b with an exchange of the xand yregisters. The task of the d-ary Deutsch-Jozsa algorithm is to determine an unknown multi-value function f: {0, 1,..., d−1} n→{0, 1,..., d−1} is either constant or balanced, while that of the d-ary Bernstein-Vazirani algorithm is to compute the close expression of a multi- value af fine function f:A0⊕A1x1...⊕Anxn, using only a single call of quantum oracle. When dequals to 2, the two algorithms return to the original Deutsch ’s algorithms. The key part is the implementation of f(x)⊕dyby the MVCU gate. The outcome of the algorithms is measured in the computation basis of the x-register states. b–iMeasured probability distributions (normalised coincidence counts) of the x-register in the computational basis. Results in b–hdemonstrate that the d-ary Deutsch- Jozsa algorithm allows the determination of whether f(x) is constant ( b)o r balanced ( c–h). Results in b,c,i,hshow the d-ary Bernstein-Vazirani algorithm can determine the expression of af fine functions f:b,f(x)i s constant and A1=0;c,f(x)i sa f fine and A1=1;i,f(x)i sa f fine and A1=2; h,f(x)i sa f fine and A1=3; Dotted boxes in (b--i) refer to theoretical probability distributions. Experimental probability distributions (coloured bars) are obtained from photon coincidences, which are accumulated by 20s per measurement. The classical fidelity Fcpresents the success probability of each measurement. In order to make the small error bars visible in the plots, they are plot by ± 3 σ. The values in parentheses refer to ± 1 σuncertainty. All error bars are estimated from photon Poissonian statistics.ARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-022-28767-x 6 NATURE COMMUNICATIONS | (2022) 13:1166 | https://doi.org/10.1038/s41467-022-28767-x | www.nature.com/naturecommunications and a randomised gate Urandom (see their explicit forms in Supplementary). Each pie chart presents one dit measurement outcomes, and the area of each coloured sector denotes measured probability distributions in the computational basis of {0ji,1ji,2ji,3ji}, respectively. In Fig. 5b, c, the eigenphases of Z4 andF4gates can be described by a single dit. Figure 5ds h o w st h e computed eigenphases of the Urandom gate with an accuracy of 12 dits, by running the algorithm with a number of 12 interactions on thed- Q P U .I n s t e a d ,i nt h eq u b i t - b a s e dd e v i c e ,a c h i e v i n gt h es a m ecomputational accuracy of ± 4−12requires a number of 24 computational interactions. And t he achieved computational accura- cies of 12 quarts are suf ficient for
Question 55multiple-choice
Quantum Markov semigroups (QMS) are fundamental tools in the study of open quantum systems and their dynamics, often analyzed using advanced functional analytic and operator algebra techniques. The interplay between group symmetries and non-commutative functional spaces is central to transferring classical analytical results to the quantum domain.
Which operator space is specifically required when estimating properties of non-primitive quantum Markov semigroups with non-trivial fixed-point algebras, rather than relying solely on Schatten Lp spaces?
1) Hilbert-Schmidt spaces
2) Trace-class operator spaces
3) Unconditioned Lp spaces
4) Conditioned or amalgamated Lp spaces
5) Banach lattice spaces
6) Self-adjoint operator spaces
7) Classical L∞ spaces
✓ Correct Answer:
The correct answer is 4) Conditioned or amalgamated Lp spaces.
📚 Reference Text:
II.8.Unfortunately, it is not the decoherence time or any other interesting quantity for Litself which transfers to all the Ln, but the underlying group which gives the corresponding estimates. Thus, the choice of the group and the classical Markov semigroup on it are particularly important. III. N ONCOMMUTATIVE LpSPACES AND NORM TRANSFERENCE In this section we introduce the main conceptual ideas of this article that we called group transference techniques . These ideas and subsequent mathematical results are mostly contained in [6]. All the applications we study in this article are concerned with the properties of certain (non-commutative) functional Lp spaces. When studying primitive QMS, only the usual (normalized Schatten) Lpspaces are required. However, in our case we are interested in non-primitive QMS withnon-trivial fixed-point algebra. As first illustrated in [16], the relevant Lpspaces in this case are the conditioned or amalgamated Lpspaces. Furthermore, the transference techniques require to look at the amplification of the classical semigroup (St)t≥0to the algebra L∞(G)⊗ B(H)(see Lemma II.1). This in turn makes it necessary to consider completely bounded version of the Lp spaces. All these notions are introduced in Section III-A. Section III-B is dedicated to the presentation of the transference techniques. We present them in a general framework, as we believe they can also be useful in other settings (see [6] for an other example of application in quantum information theory). Finally we specialize to QMS in Section III-C, where these transference techniques are applied to transfer estimates on the classical Markov kernel to the QMS. A.Lpnorms and entropies We are now going to introduce several Lpnorms and entropies related to von Neumann algebras. Although this may not be clear at first sight, it turns out that many of them are just the sandwiched Rényi entropies [17], [18] in disguise, as we will clarify. In the following M is a finite von Neumann algebra and ∶M→Ca normalized normal, faithful, tracial state (i.e. (IH)=1). Let us recall the definition of the noncommutative Lp spaces via (15) /divides.alt0x/parallel.alt1Lp( )∶=[ (/divides.alt0x/divides.alt0p)]1/slash.leftp: (16) ThenLp(M; )≡Lp(M)is the completion of Mwith respect to this norm. Indeed, for 1≤p≤∞the space is a Banach space such that Lp(M; )∗=L^p(M; )holds for1 p+1 ^p=1and1≤p<∞. In this article we will focus on three types of von Neumann algebras: ●Our main example is M=Mm, the space of m×m matrices over the field of complex numbers, and (x)≡ m(x)≡1 mTr(x). To keep the notations at a more abstract level, we shall most of the time refer to a finite dimensional Hilbert space Hand to the algebra of (bounded) linear operators B(H) and we denote by Lp(B(H))the corresponding non-commutative Lpspace. ●If(E;F;)is a probability space, where Fis a-algebra on the set Eanda probability distribution, then the set of bounded complex-valued functionM=L∞()is a von-Neumann algebra, ∶f/uni21A6E(f)is a normal, faithful and tracial state and the corresponding Lpspaces are the usual Lp(). ●The last key example in the transference principle is the algebra of bounded Mm-valued function on a probability space (E;F;),M=L∞(E;Mm), with trace given by ∶f/uni21A6/integral.dispE m(f(x))d(x): 7 For
Question 56multiple-choice
Quantum attacks on code-based cryptosystems often leverage the Hidden Subgroup Problem (HSP) and strong Fourier sampling techniques. Understanding subgroup structure in permutation and linear groups is crucial for analyzing the quantum security of systems like McEliece.
Which property of the subgroup relevant to McEliece-type cryptosystems ensures its indistinguishability by strong quantum Fourier sampling and thus underpins quantum resistance against HSP-based attacks?
1) Having both large size and large minimal degree relative to n and log n
2) Being abelian and cyclic
3) Consisting only of the identity permutation
4) Having a small automorphism group but a large scrambler group
5) Containing all possible permutations of code positions
6) Being generated by elements of order two
7) Having a determinant equal to one for all group members
✓ Correct Answer:
The correct answer is 1) Having both large size and large minimal degree relative to n and log n.
Topological quantum computing utilizes the braiding of anyons to perform fault-tolerant quantum operations, with algebraic structures like braid groups and the Temperley-Lieb algebra playing a central role in gate construction and entanglement generation. The ability to directly create multipartite entangled states enhances the efficiency of quantum circuit design.
Which of the following statements best describes the significance of constructing braiding operators based on the Temperley-Lieb algebra for multi-qubit quantum systems?
1) They restrict quantum gate implementation to only two-qubit operations.
2) They eliminate the use of single-qubit gates in entanglement generation.
3) They generalize braiding operations to enable direct creation of multi-qubit entangled states from separable bases.
4) They decrease the fault-tolerance of topological quantum computers.
5) They replace the braid group theory with a purely statistical mechanical approach.
6) They are limited to generating only Bell states, not GHZ or cluster states.
7) They require classical information storage for maintaining topological protection.
✓ Correct Answer:
The correct answer is 3) They generalize braiding operations to enable direct creation of multi-qubit entangled states from separable bases..
📚 Reference Text:
Title: Quantum entanglement, unitary braid representation and Temperley-Lieb algebra Year: 2010 Paper ID: c2d69aeb4f9db4c24d6b0f1c6a1dba78bb9f8c28 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/c2d69aeb4f9db4c24d6b0f1c6a1dba78bb9f8c28 Abstract: Important developments in fault-tolerant quantum computation using the braiding of anyons have placed the theory of braid groups at the very foundation of topological quantum computing. Furthermore, the realization by Kauffman and Lomonaco that a specific braiding operator from the solution of the Yang-Baxter equation, namely the Bell matrix, is universal implies that in principle all quantum gates can be constructed from braiding operators together with single qubit gates. In this paper we present a new class of braiding operators from the Temperley-Lieb algebra that generalizes the Bell matrix to multi-qubit systems, thus unifying the Hadamard and Bell matrices within the same framework. Unlike previous braiding operators, these new operators generate directly, from separable basis states, important entangled states such as the generalized Greenberger-Horne-Zeilinger states, cluster-like states, and other states with varying degrees of entanglement.
Question 58multiple-choice
Quantum computing leverages principles from quantum mechanics to process information in ways fundamentally different from classical computing. Understanding the roles of quantum gates and physical implementation challenges is key to advancing the field.
Which experimental setup is commonly used to demonstrate quantum interference and implement quantum logic gates using photon polarization?
1) Josephson junction
2) Mach-Zehnder interferometer
3) Bell test apparatus
4) Quantum dot array
5) Ion trap
6) Fabry-Pérot cavity
7) SQUID (Superconducting Quantum Interference Device)
✓ Correct Answer:
The correct answer is 2) Mach-Zehnder interferometer.
📚 Reference Text:
Title: Physical foundations of quantum informatics: from quantum mechanics through quantum computing to quantum cryptography Year: 2022 Paper ID: f56fde97fd1fee4a8d98973ec25145cb2d90dc7e Source: semantic-scholar URL: https://www.semanticscholar.org/paper/f56fde97fd1fee4a8d98973ec25145cb2d90dc7e Abstract: A methodical analysis of the basic problem related to quantum calculations of parameters of physical systems was made. Emphasis is placed on the physical principles of the operation of a quantum computer, with an emphasis on the fact that simultaneous access to all quantum states is important in quantum computing, which allows the simultaneous change of the quantum state from all superpositions in the qubit system. Emphasis is placed on the fact that in quantum algorithms the Fourier transform and the Hadamard transform are the basic operations - as a simple discrete Fourier transform. The reader's attention is drawn to the fact that quantum computing is primarily implemented in quantum objects with the properties of elementary NOT gates and controlled CNOT, which can be implemented on a Mach-Zehnder interferometer using the phenomena of photon interference and rotation of its polarization vector. Despite the progress of conventional computers, the need for the development of quantum computing is due to the technological limitation due to the dimensional quantization of the electronic spectrum and the exponential increase in the time of calculations by classical algorithms when the volume of data increases. However, the widespread use of quantum computers is limited by a number of problems. This is, first of all, insufficient accuracy and high sensitivity to external influences that can destroy the quantum state. Therefore, to increase the accuracy of calculations on a quantum computer, the calculation algorithm must be repeated a certain number of times, and to avoid the destruction of the quantum states of the qubit, low temperatures are used.
Question 59multiple-choice
The quantum Fourier transform (QFT) is an essential operation in quantum computing, especially for algorithms that exploit the structure of cyclic groups. Understanding performance bounds and resource requirements for QFT is crucial for quantum algorithm design and practical implementation.
Which statement most accurately describes a key advantage of having explicit bounds on the number of qubits required for a given tolerance in quantum Fourier transform algorithms over cyclic groups?
1) It allows the QFT to be replaced with classical algorithms in all cases.
2) It ensures that the QFT can be performed without any noise or errors.
3) It guarantees exponential speedup for all problems using QFT.
4) It removes the necessity for simulation before hardware deployment.
5) It enables heuristic estimation of resource requirements rather than exact quantification.
6) It allows researchers and practitioners to precisely optimize quantum hardware usage for desired algorithmic precision.
7) It restricts the applicability of QFT to only large quantum computers.
✓ Correct Answer:
The correct answer is 6) It allows researchers and practitioners to precisely optimize quantum hardware usage for desired algorithmic precision..
📚 Reference Text:
Title: A quantum Fourier transform algorithm Year: 2004 Paper ID: 38a50c30456e53cc55a161106d0da5775d023f2d Source: semantic-scholar URL: https://www.semanticscholar.org/paper/38a50c30456e53cc55a161106d0da5775d023f2d Abstract: Algorithms to compute the quantum Fourier transform over a cyclic group are fundamental to many quantum algorithms. This paper describes such an algorithm and gives a proof of its correctness, tightening some claimed performance bounds given earlier. Exact bounds are given for the number of qubits needed to achieve a desired tolerance, allowing simulation of the algorithm.
Question 60multiple-choice
In graph theory and quantum computing, decomposing complex graphs into simpler structures like cliques allows for efficient analysis of Hamiltonians that govern quantum interactions. Specialized algorithms use hierarchical tree structures to systematically break down these graphs.
Which statement best describes the recursive rule for computing the Hamiltonian at a vertex in a tree-clique decomposition of a graph?
1) The Hamiltonian is equal to the sum of Hamiltonians of its children plus the edge Hamiltonians incident to the vertex.
2) The Hamiltonian is obtained by multiplying the complete subgraph Hamiltonian by the number of tree leaves below the vertex.
3) The Hamiltonian is given by the union of Hamiltonians from all ancestors of the vertex in the decomposition tree.
4) The Hamiltonian excludes all interactions involving vertices outside the immediate subgraph of the node.
5) The Hamiltonian at each vertex is computed by taking the intersection of clique Hamiltonians from its children.
6) The Hamiltonian equals the complete graph Hamiltonian on the vertex's associated subgraph minus the sum of Hamiltonians of its children.
7) The Hamiltonian for each node is constant regardless of its position in the tree.
✓ Correct Answer:
The correct answer is 6) The Hamiltonian equals the complete graph Hamiltonian on the vertex's associated subgraph minus the sum of Hamiltonians of its children..
📚 Reference Text:
of the algorithm first note that an n-vertex graph can be decomposed into its connected components in time O(n2) (using depth first or breadth first search). This is the dominant step in the inner loop of the algorithm, so the algorithm takes time at most O(|V(G(w))|2) to complete the inner loop that removes vertex wfrom activeTleaves . Next, note that every vertex in T Accepted in Quantum2024-03-25, click title to verify. Published under CC-BY 4.0. 52 appears in activeTleaves at most once. Then, letting ddenote the depth of T, the total runtime of the algorithm is bounded by: O /summationdisplay w∈V(T)|V(G(w))|2 =O d/summationdisplay i=1/summationdisplay w:depth(w)=i|V(G(w))|2 (6.16) ≤O/parenleftiggd/summationdisplay i=1n2/parenrightigg ≤O(n3), (6.17) where the first inequality follows from Item (2) of Theorem 6.3 and the second inequality follows from Item (1). 6.2.2 Clique decompositions and Hamiltonians Finally we show how the tree graph decomposition can be used to write the QMC Hamil- tonian associated with a graph Gas a signed sum of clique Hamiltonians and smaller “residual” graph Hamiltonians. Theorem 6.5. LetGbe a graph and T(G) ={T,{G(v1),...,G (vn)}}be the tree-clique decomposition of G. Also, for any graph G, letK(G)denote the complete graph on the vertices ofG. Then the following claims hold: (1)For any vertex v∈Twith children c1,...,ckwe have HG(v)=HK(G(v))−/summationdisplay j∈{1,...,k}HG(cj) (6.18) (2)LetLdenote the set of leaf vertices in T, andRbe all non-leaf vertices. Also, for any vertex v∈T, letd(v)denote the depth of vertex vin the the tree, with the root vertex having depth d(v1) = 0 . Then we have HG=/summationdisplay r∈R(−1)d(r)HK(G(r))+/summationdisplay l∈L(−1)d(l)HG(l) (6.19) Proof. Part (1) of the theorem follows from Item (2) in Definition 6.2 which gives G(v)c=/uniondisplay j∈{1,...,k}G(cj) (6.20) and hence HG(v)=HK(G(v))−HG(v)c=HK(G(v))−/summationdisplay j∈{1,...,k}HG(cj). (6.21) Part (2) then follows from repeated application of (1), beginning with the root vertex ofT. Accepted in Quantum2024-03-25, click title to verify. Published under CC-BY 4.0. 53 6.2.3 Example of clique decomposition, Algorithm 6 Before giving an explicit example of Algorithm 6 we note any complete k-partite graph has a particularly simple tree clique decomposition which Algorithm 6 finds in one step. This is because a complete k-partite graph is by definition the complement of the disjoint union ofkdifferent complete graphs; these are the output of the algorithm. Now we give an explicit illustration of Algorithm 6 on a graph which is not a complete k-partite graph. 78 9103 2 1 6 54 Figure 1: This is the input graph to the algorithm. The outputtree Tis in Section 6.2.3 and the graphs G(vj) are in Figure 3. v1 v2 v5 v3 v4 Figure 2: This is the output Tfrom Algorithm 6 run on the graph in Figure 1. Accepted in Quantum2024-03-25, click title to verify. Published under CC-BY 4.0. 54 78 9103 2 1 6 54 G(v1)78 93 2 1 6 54 G(v2)3 2 1 6 54 G(v3)77 78 9 G(v4)77 10 G(v5) Figure 3: This is the list of graphs which are output. The leaf graph, the graphs on which the algorithm terminates, correspond to v3,v4, andv5. The first of these is a
Question 61multiple-choice
Matrix product states (MPS) are powerful computational tools for simulating quantum many-body systems and have been increasingly applied to lattice gauge theories. In these simulations, truncating the Hilbert space to a finite set of gauge group representations is a crucial practical step.
In the context of simulating the Schwinger model with MPS methods under a uniform electric background field, why is it valid to truncate the Hilbert space of gauge fields to a finite number of irreducible representations?
1) The weight of each representation in the ground state decays exponentially with its quadratic Casimir invariant, making higher representations negligible.
2) Only the lowest representation contributes to the physical properties of the model.
3) All gauge group representations have equal contribution in the ground state, making truncation arbitrary.
4) The gauge fields in the Schwinger model are inherently finite-dimensional.
5) Truncation is always valid for any gauge theory regardless of the background field.
6) The electric background field forces all higher representations to vanish.
7) The single-particle spectrum remains unchanged even without truncation.
✓ Correct Answer:
The correct answer is 1) The weight of each representation in the ground state decays exponentially with its quadratic Casimir invariant, making higher representations negligible..
📚 Reference Text:
Title: Finite-representation approximation of lattice gauge theories at the continuum limit with tensor networks Year: 2017 Paper ID: 1df78f472504e50ae8168ac6d665668566f4418f Source: semantic-scholar URL: https://www.semanticscholar.org/paper/1df78f472504e50ae8168ac6d665668566f4418f Abstract: It has been established that matrix product states can be used to compute the ground state and single-particle excitations and their properties of lattice gauge theories at the continuum limit. However, by construction, in this formalism the Hilbert space of the gauge fields is truncated to a finite number of irreducible representations of the gauge group. We investigate quantitatively the influence of the truncation of the infinite number of representations in the Schwinger model, one-flavor QED 2, with a uniform electric background field. We compute the two-site reduced density matrix of the ground state and the weight of each of the representations. We find that this weight decays exponentially with the quadratic Casimir invariant of the representation which justifies the approach of truncating the Hilbert space of the gauge fields. Finally, we compute the single-particle spectrum of the model as a function of the electric background field.
Question 62multiple-choice
Quantum groups generalize classical Lie groups by introducing a deformation parameter q, leading to non-commutative algebraic structures and novel representation theory. The quantum universal enveloping algebra Uq(SL(n)) and related constructs play a central role in mathematical physics and non-commutative geometry.
Which of the following statements about the quantum determinant and its role in the structure of SLq(2) is correct?
1) The quantum determinant in SLq(2) is defined identically to the classical determinant, without q-dependent coefficients.
2) In SLq(2), the matrix constructed from τF and its derivatives always belongs to the quantum group GLq(2).
3) The quantum determinant det_q always commutes with all elements of SLq(2).
4) The classical group identities (g-identities) are directly applicable without adjustment in the quantum group setting.
5) The quantum determinant for SLq(2) does not involve operator ordering or q-dependent terms.
6) The quantum determinant det_q in SLq(2) is formulated using q-dependent coefficients and specific operator ordering to maintain invariance in the non-commutative setting.
7) Matrices constructed from τF and its derivatives in SLq(2) always satisfy commutative multiplication properties.
✓ Correct Answer:
The correct answer is 6) The quantum determinant det_q in SLq(2) is formulated using q-dependent coefficients and specific operator ordering to maintain invariance in the non-commutative setting..
📚 Reference Text:
restrict ourselves to the case of G=SLq(2) and introduce: H1 1=τF=a+b¯t+ct+dt¯t. (98) If H1 2=D¯tH1 1=b+dt, H2 1=DtH1 1=c+d¯t, H2 2=D¯tDtH1 1=d,(99) we see that Ha bis actually not lying in GLq(2) (for example, H1 2H2 1/ne}ationslash=H2 1H1 2), i.e. a matrix consisting of the τFand its derivatives, despite these are all elements of A(G), does not longer belong toGq. Thus, it is not reasonable to consider det qH(or the definition of Hshould be somehow modified). Instead the appropriate formula for the case ofSLq(2) looks like τF(2)= detqg=H1 1H2 2−qH1 2M− ¯tH2 1=τFDtD¯tτF−qD¯tτFM− tDtτF.(100) We shall not go into more discussion of transition from g-identity to H-identities, because it involves some art in the work with appropriate time-variables, and is not yet brought to a reasonably simple form. Instead we present a few more formulas, w hich can be illuminating for some readers. 5.3 Comments on the quantum case The first thing we wish to give some more details on is the statement (9 3). The Lie algebra SL(n) is generated by operators T±αand Cartan operators Hβ, such that [Hβ,T±α] =±1 2(αβ)T±α. All elements of all representations are eigenfunctions of Hβ, 26 Hβ|λ/an}bracketri}ht=1 2(βλ)|λ/an}bracketri}ht. The highest weight of representation F(k)isµk. Vectors µk’s are “dual” to thesimplerootsαi,i= 1,...,r: (µiαj) =δij, andρ=1 2/summationtext α>0α=/summationtext iµi. Representation F(1)consists of the states ψi=T−(i−1)...T−2T−1ψ1, i= 1,...,n. (101) Moreover T−iψj=δijψi+1 (102) (thus, forT−=/summationtextr i=1T−i Ti −ψj=ψj+iand (60) follows), and λ(ψi) =µ1−α1−...−αi1. (103) HereT±i≡T±αiare generators, associated with the simple roots. Let us denote t he corre- sponding basis in Cartan algebra Hi=Hαi, andHi|λ/an}bracketri}ht=1 2(αiλ)|λ/an}bracketri}ht=λi|λ/an}bracketri}ht. Then λ(j) i≡λi(ψj) =1 2(δij−δi,j−1). (104) This formula, together with (101) and (105) implies that ||ψi||2= 1, and, since comultiplica- tion formula in the classical case is just ∆( T) =T⊗I+I⊗T, it is obvious that ψ[1...ψk]are all highest weight vectors (i.e are annihilated by all ∆ k(T+i) and, thus by all the ∆ k(T+α)). Quantum universal enveloping algebra Uq(SL(n)) is generated by T±iandq±Hiwith basic commutation relations ( aijis the Cartan matrix of SL(n)) qHiT±jq−Hi=q±aijT±j, [T+i,T−j] =δijq2Hi−q−2Hi q−q−1(105) and comultiplication law ∆(T±i) =qHi⊗T±i+T±i⊗q−Hi, ∆(q±Hi) =q±Hi⊗q±Hi.(106) Comultiplication formulas for T±αin the case of non-simple roots are corollaries of these and look more sophisticated. For example, for the “height 2” α, such that T±α=±[T±αi,T±αi+1], 27 we have ∆(T−α) =−[∆(Tαi),∆(Tαi+1)] =qHα⊗Tα+Tα⊗q−Hα+ +(q1/2−q−1/2)/bracketleftig (T−αi⊗T−αi+1)(qHi+1⊗q−Hi)−(T−αi+1⊗T−αi)(qHi⊗q−Hi+1)/bracketrightig .(107) Given multiplication formulas, one can easily check that indeed (93) is t rue. For example, forF(2): ∆(T+i)(ψ1ψ2−qψ2ψ1) =δi,1(qλ(1) 1ψ1ψ1−q1−λ(1) 1ψ1ψ1) = 0, (108) becauseλ(1) 1=1 2. Thus Ψ(2) 12≡ψ[1ψ2]qis indeed the highest weight vector. Similarly ( i
Question 63multiple-choice
The Yang-Baxter equation plays a central role in mathematical physics and group theory, with set-theoretic solutions linked to specific algebraic structures. Classifying and constructing these solutions involves interpreting connections between groups and permutation symmetries.
Which property precisely characterizes all involutive Yang-Baxter groups (IYB groups) identified through the correspondence with involutive non-degenerate set-theoretic solutions to the Yang-Baxter equation?
1) They are always simple non-abelian groups.
2) They are necessarily nilpotent groups of class one.
3) They contain a unique subgroup isomorphic to the free abelian group Faₙ.
4) They are precisely cyclic groups of prime order.
5) They must be infinite and non-solvable.
6) They are defined only for abelian groups.
7) They are all solvable groups.
✓ Correct Answer:
The correct answer is 7) They are all solvable groups..
📚 Reference Text:
Title: Involutive Yang-Baxter groups Year: 2008 Paper ID: 5b3d75eaec33a163cd2da04b967099103f8232e2 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/5b3d75eaec33a163cd2da04b967099103f8232e2 Abstract: In 1992 Drinfeld posed the question of finding the set-theoretic solutions of the Yang-Baxter equation. Recently, Gateva-Ivanova and Van den Bergh and Etingof, Schedler and Soloviev have shown a group-theoretical interpretation of involutive non-degenerate solutions. Namely, there is a one-to-one correspondence between involutive non-degenerate solutions on finite sets and groups of I-type. A group G of I-type is a group isomorphic to a subgroup of Fa n ⋊ Sym n so that the projection onto the first component is a bijective map, where Fa n is the free abelian group of rank n and Sym n is the symmetric group of degree n. The projection of G onto the second component Sym n we call an involutive Yang-Baxter group (IYB group). This suggests the following strategy to attack Drinfeld's problem for involutive non-degenerate set-theoretic solutions. First classify the IYB groups and second, for a given IYB group G, classify the groups of I-type with G as associated IYB group. It is known that every IYB group is solvable. In this paper some results supporting the converse of this property are obtained. More precisely, we show that some classes of groups are IYB groups. We also give a non-obvious method to construct infinitely many groups of I-type (and hence infinitely many involutive non-degenerate set-theoretic solutions of the Yang-Baxter equation) with a prescribed associated IYB group.
Question 64multiple-choice
In multipartite quantum systems, the classification and construction of entangled states under local operations is fundamental for quantum information processing. Locally maximally entangled (LME) states, which generalize Bell states, play a crucial role in this domain.
Which defining property distinguishes a locally maximally entangled (LME) state in a composite quantum system?
1) Each subsystem's reduced density matrix is proportional to the identity operator.
2) The global state is invariant under all local unitary transformations.
3) The state can be transformed into any other state by local operations and classical communication.
4) Each subsystem is in a pure quantum state.
5) The state maximizes the global entropy of the composite system.
6) The state is an eigenstate of the total Hamiltonian.
7) Each subsystem's reduced density matrix is diagonal with non-equal eigenvalues.
✓ Correct Answer:
The correct answer is 1) Each subsystem's reduced density matrix is proportional to the identity operator..
📚 Reference Text:
. . . . . 23 4.1.1 Entanglement structure and the action of K= SU(d1)×···×SU(dn) 23 4.1.2 SLOCC orbits and the action of G= SL(d1,C)×···×SL(dn,C) . . . 23 4.2SLME/Kas a symplectic manifold . . . . . . . . . . . . . . . . . . . . . . . . 24 4.3SLME/Kas a complex manifold . . . . . . . . . . . . . . . . . . . . . . . . . 25 4.4 Gradient flow to LME states . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 7 4.5 Algebraic characterization of P(H)//G. . . . . . . . . . . . . . . . . . . . . 28 5 Understanding SLMEusing geometric invariant theory 31 5.1 Proof of theorems governing existence and dimension of SLME. . . . . . . . 33 5.2 Explicit results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 1 A Explicit construction of all LME states for (d1,d2,d3) = (2,B,C) 39 A.1 Sudoku States for n= 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 B Representation theory construction of LME states for (d1,d2,d3) = (2,p,p) with prime p 44 B.1 Irreducible representations of UT(3,p) . . . . . . . . . . . . . . . . . . . . . 44 B.2 The group H=UT(3,p)⋊φZ2. . . . . . . . . . . . . . . . . . . . . . . . . 45 B.3 Irreducible representations of UT(3,p)⋊Z2. . . . . . . . . . . . . . . . . . 47 C Multipart systems for which the generic state has trivial s tabilizer 49 1 Introduction Consider a multipart quantum system whose pure states are vecto rs in a tensor product Hilbert space H=H1⊗···⊗H n with subsystems Hiof dimension di. We will say that a state in His locally maximally entangled (LME) if the reduced density matrix corresponding to ea ch elementary subsystem is a scalar multiple of the identity operator on that subsystem.1Fixing the dimension vector (d1,...,dn), we define SLME⊂ Hto be the subset of states which are locally maximally entangled: SLME=/braceleftbigg |Ψ/an}b∇acket∇i}ht ∈ H/vextendsingle/vextendsingle/vextendsingleρi≡tr¯i|Ψ/an}b∇acket∇i}ht/an}b∇acketle{tΨ|=1 di1 1/bracerightbigg . (1.1) LMEstates play animportant roleinmany applicationsof quantum mec hanics andquantum information theory. Their importance has been pointed out by many authors in the past, for example [Kly02, VDD03, Sc04]. Well-known examples of LME states include Bell states |Ψ/an}b∇acket∇i}ht=1√ d/summationdisplay i|i/an}b∇acket∇i}ht⊗|i/an}b∇acket∇i}ht ∈ Hd⊗Hd, (1.2) 1Such states are sometimes
Question 65multiple-choice
Infinite abelian group theory explores properties of groups such as slenderness, torsion, and divisibility, with important connections to algebraic topology and set theory. The Baer–Specker group and p-adic numbers play central roles in classifying torsion-free abelian groups of rank one.
Which criterion precisely characterizes a torsion-free abelian group of cardinality less than the continuum as slender?
1) It has only finite-rank free subgroups.
2) It has no non-trivial divisible subgroups.
3) It is isomorphic to a subgroup of the p-adic integers for some prime p.
4) Every homomorphism from the Baer–Specker group into it is injective.
5) It contains a copy of the Baer–Specker group as a subgroup.
6) It is generated by finitely many elements.
7) Its endomorphism ring is a discrete valuation ring.
✓ Correct Answer:
The correct answer is 2) It has no non-trivial divisible subgroups..
📚 Reference Text:
Title: Fuchs' Problem When Torsion-Free Abelian Rank-One Groups are Slender Year: 2009 Paper ID: e85f50983e13773ff9ba461cc52581ad18aad9be Source: semantic-scholar URL: https://www.semanticscholar.org/paper/e85f50983e13773ff9ba461cc52581ad18aad9be Abstract: We combine Baer’s classification in [Duke Math. J. 3 (1937), 68–122] of torsion-free abelian groups of rank one together with elementary properties of p-adic numbers to give a new solution to research problem 26 posed by Fuchs in his book on abelian groups in 1958. Every group considered in this paper is abelian with its group law written additively. The Baer–Specker group Π = Zא0 is the direct product of countably many copies of Z. Its extensive study in the literature has given rise to a wealth of interesting problems and a number of unexpected connections with diverse mathematical areas such as homology theories in algebraic topology, infinitary logic and various aspects of set theory. We refer the reader to [2], [3], [4], [6] and [11] for good accounts of some of these connections. For a positive integer n, let en denote the element of Π whose n-th coordinate equals 1 and all its other coordinates equal 0. Following Łoś, a torsion-free group G is called slender if for every homomorphism φ : Π → G we have φ(en) = 0, for all but finitely many n. Specker ([10]) showed that Z is slender. Research problem 26 in Fuchs’ book (page 184 in [5]) reads as follows: Problem. Find all slender groups of rank one. A partial answer to the problem was given by Łoś (see Theorem 47.3 in [5]). The problem was fully solved by Sąsiada ([9]) who in fact showed that: Theorem 1. A torsion-free group of cardinality less than 2א0 is slender if and only if it has no non-trivial divisible subgroups. 2000 Mathematics Subject Classification. Primary 20K15; Secondary 20K25.
Question 66multiple-choice
Quantum computing devices often impose restrictions on which qubits can directly interact, necessitating circuit optimizations for efficient implementation of algorithms. The Quantum Fourier Transform (QFT) is a core subroutine in many quantum algorithms, and its gate count is a crucial factor on noisy intermediate-scale quantum (NISQ) hardware.
When designing QFT circuits for quantum computers with linear nearest-neighbor connectivity, which outcome is most directly achieved by optimizing the circuit for the hardware's connectivity constraints?
1) Increased robustness against decoherence by using more Toffoli gates
2) Enhanced algorithmic speed by parallelizing all one-qubit gates
3) Higher accuracy in phase estimation due to longer circuit depth
4) Substantial reduction in the total number of required CNOT gates
5) Improved entanglement of non-adjacent qubits without additional gates
6) Elimination of noise in single-qubit measurement operations
7) Automatic error correction of all two-qubit gates
✓ Correct Answer:
The correct answer is 4) Substantial reduction in the total number of required CNOT gates.
📚 Reference Text:
a transpiler23 to transform an input circuit into a circuit that satisfies the specific NN con- dition, which is required in each IBM quantum computer. In this section, we put our QFT circuits and the conventional QFT circuits (such as the circuit in Fig. 1) in the Qiskit transpiler for implementation on IBM quantum computers: (1) IBM_Nairobi, a 7-qubit quantum computer using the Falcon r5.11H processor, (2) IBMQ_Guadalupe, a 16-qubit quantum computer using the Falcon r4P processor, (3) IBM_Cairo, a 27-qubit quantum computer using the Falcon r5.11 processor, and (4) IBM_Washington, a 127-qubit quantum computer using the Eagle r1 processor10. We transpiled 3- to 7-qubit QFT on the IBM_Nairobi, 3- to 16-qubit QFT on the IBMQ_Guadalupe, 3- to 27-qubit QFT on the IBMQ_Cairo, and 3- to 127-qubit QFT on the IBM_Washington. Each QFT circuit is transpiled 100 times. Next, we chose the minimal number of CNOT gates required to syn - thesize the QFT and compared them. As a result, we confirmed that using our QFT circuit as input requires fewer CNOT gates than using the conventional QFT circuit for all cases. The results can be found in Fig. 8. Implementation of QFT on actual quantum hardware. We implemented QPE using a 3-qubit QFT on the IBM_Nairobi10 and the Rigetti-Aspen-1111, a 40-qubit superconducting quantum computer, to compare their performance. The connectivity between qubits used for the implementation of QPE can be found in Fig. 7. QPE is an algorithm for finding an eigenvalue of a unitary operator using a corresponding eigenstate and QFT. A brief explanation of QPE can be found in the “ Background ” section. In this study, we chose the unitary operator U and the corresponding eigenvector |u� as follows: We chose θ as 1/8, 2/8, 3/8, …, and 7/8. The QPE circuits are synthesized using our method. If we use a quantum computer without noise when implementing QPE, we can get the right results with one execution for each θ . However, the quantum computers we used are noisy. Therefore, we implemented QPE 1000 times for each θ on each quantum computer.(11) U=/parenleftbigg10 0e2πiθ/parenrightbigg ,|u�=/parenleftbigg 0 1/parenrightbiggTable 1. The number of CNOT gates in QFT circuits for LNN architecture. The first column represents the number of qubits in the QFT circuit, the second column represents our results, the third to the fifth columns represent the results of previous studies14,17,18, and the sixth column represents the improvement rate of our circuit compared to the best-known result.Ours Ref.14Ref.17Ref.18Improvement (%) n n2 + n − 4 (5/2)(n2 − 1) – – ~ 60 5 26 50 31 – ~ 16.13 6 38 75 48 – ~ 20.83 7 52 105 105 75 ~ 30.67 8 68 140 124 121 ~ 43.80 9 86 180 192 165 ~ 47.88 10 106 225 240 225 ~ 52.89 Figure 7. Qubit connectivity of quantum devices (a ) Circuit diagram of IBM_Nairobi10, showing the connectivity of qubits. Qubits labeled 1, 3, 5, and 4 are used to implement QPE. (b ) Partial circuit
Question 67multiple-choice
In quantum field theory, the interaction between Dirac fermions and non-Abelian monopoles gives rise to novel symmetry structures and operator properties. The mathematical framework often involves complex groups and specialized operators that influence the classification of quantum states.
Which statement best characterizes the role of the matrix F when the parameter A is real in a system with Dirac fermion doublets and non-Abelian monopole potential?
1) F generates time-reversal symmetry and always leaves the Hamiltonian invariant.
2) F belongs to SU(2) and acts as a symmetry generator for the Hamiltonian by transforming basis wave functions.
3) F is a projection operator onto orthogonal quantum states for all values of A.
4) F represents charge conjugation and maps fermions to antifermions exclusively.
5) F becomes non-unitary and fails to preserve inner products when A is real.
6) F implements parity inversion and does not affect the Hilbert space structure.
7) F diagonalizes the N operator for both real and complex values of A.
✓ Correct Answer:
The correct answer is 2) F belongs to SU(2) and acts as a symmetry generator for the Hamiltonian by transforming basis wave functions..
📚 Reference Text:
Title: The doublet of Dirac fermions in the field of the non-Abelian monopole, isotopic chiral symmetry, and parity selection rules Year: 1999 Paper ID: fb43e6df11fa1aa41079f15c58b1f298235ac59e Source: semantic-scholar URL: https://www.semanticscholar.org/paper/fb43e6df11fa1aa41079f15c58b1f298235ac59e Abstract: The paper concerns a problem of the Dirac fermion doublet in the external monopole potential obtained by embedding the Abelian monopole solution in the non-Abelian scheme. In this case, the doublet-monopole Hamiltonian is invariant under operations consisting of a complex and one parametric Abelian subgroup in S0(3.C). This symmetry results in a certain freedom in choosing a discrete operator N(A) (A is a complex number) entering the complete set of quantum variables. The same complex number A represents an additional parameter at the basis functions. The generalized inversion like operator N(A) affords certain generalized N(A)-parity selection rules. All the different sets of basis functions Psi(A) determine the same Hilbert space. The functions Psi(A) decompose into linear combinations of Psi(A=0): Psi(A) = F(A) Psi(A=0). However, the bases considered turn out to be nonorthogonal ones when A is a complex number; the latter correlates with the non-self-conjugacy of the N(A) at complex A-s. The meaning of possibility to violate the quantum-mechanical regulation on self-conjugacy as regards the operator N(A) is discussed. Also, the problem of possible physical status for the matrix F(A) at real A-s is considered in full detail: since the matrix belongs formally to the gauge group SU(2), but being a symmetry for the Hamiltonian this F(A) generates linear transformations on basis wave functions.
Question 68multiple-choice
Quantum annealing and Ising Hamiltonian models are central to applying quantum computing for hard optimization problems, including integer factorization and cryptanalysis. These techniques rely on encoding mathematical problems into spin systems compatible with quantum hardware.
Which process allows the mapping of a classical integer factorization problem into an Ising Hamiltonian suitable for quantum annealing, enabling solutions via quantum hardware?
1) Representing integers as binary strings and using classical brute-force search
2) Applying modular exponentiation to transform the factorization constraints
3) Renaming problem variables and mapping them to the spin domain {-1, 1}, rewriting the cost function in terms of local fields and couplings
4) Utilizing Grover's algorithm to sample possible factors
5) Encoding the problem as a quantum circuit using universal gates
6) Reducing factorization to a discrete logarithm problem in a cyclic group
7) Implementing lattice-based cryptographic schemes for enhanced security
✓ Correct Answer:
The correct answer is 3) Renaming problem variables and mapping them to the spin domain {-1, 1}, rewriting the cost function in terms of local fields and couplings.
📚 Reference Text:
respectively. Further, rename the variables p1,p2,q1,q2,c1,..., c4,t1,..., t4asv1,...,v 12, and replace vi=1−si 2(i=1,..., 12) to make the variables lie in the domain {−1,1}. Now, the above cost function fcan be rewritten as f(p1,p2,q1,q2,c1,c2,c3,c4)=2f/prime(s1,..., s12) where f/primeis given in Fig. 3. •Step 5 Now, we can review the above cost function as an Ising Hamiltonian with local fields, and the values of hiand Jijcan be derived accordingly (See Fig. 3for details). •Step 6 Solve the Ising Hamiltonian system by calling qbsolv , the Python library provided by D-Wave systems, and map the results back to the prime factorization ofN. To summary, the qubits needed for different implementations of quantum factor- ization based on QAC are shown in Table 3. Table 3 Different implementations of quantum factorization based on QAC Authors Year The largest integer can be factorized Qubits Li et al. [ 40] 2018 291,311 – Jang et al. [ 41] 2018 376,298 94 Wang et al. [ 33] 2019 1,005,973 89 Comparison of the largest integer can be factorized 123 Quantum algorithms for typical hard problems… Page 11 of 26 178 f(s1,s2,··· ,s12)=261 2s1+215 2s2+261 2s3+215 2s4−41s5−82s6+3 s7+6 s8−137 s9−81s10−107 s11−81s12 +2s1s2+7 9 s1s3+95 2s1s4−2s1s5−4s1s6−8s1s7−16s1s8−148 s1s9−84s1s10 +95 2s2s3+7 1 s2s4−8s2s5+−16s2s6+s2s7+2 s2s8+6 s2s9+6 s2s10−124 s2s11−84s2s12 +2s3s4−2s3s5−4s3s6−8s3s7−16s3s8−148 s3s9−84s3s12 −8s4s5−16s4s6+s4s7+2 s4s8+6 s4s9−84s4s10−124 s4s11+6 s4s12 +34 s5s6−4s5s7−8s5s8−8s5s9+s5s10+2 s5s11+s5s12 −8s6s7−16s6s8−16s6s9+2 s6s10+4 s6s11+2 s6s12 +34 s7s8−4s7s10−8s7s11−4s7s12 −8s8s1016s8s118s8s12 +s9s11 +794 hT=( σ(1) z,··· ,σ(12) z) = (130 .5,107 .5,130 .5,107 .5,−41,−82,3,6,−137 ,−81,−107 ,−81) J=⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝σ (1) z σ(2) z σ(3) z σ(4) z σ(5) z σ(6) z σ(7) z σ(8) z σ(9) z σ(10) z σ(11) z σ(12) z σ(1) z 02 7 9 4 7 .5 −2 −4 −8 −16 −148 −84 0 0 σ(2) z 04 7 .57 1 −8 −16 1 2 6 6 −124 −84 σ(3) z 02 −2 −4 −8 −16 −148 0 0 −84 σ(4) z 0 −8 −16 1 2 6 −84 −124 6 σ(5) z 03 4 −4 −8 −81 2 1 σ(6) z 0 −8 −16 −16 2 4 2 σ(7) z 03 4 0 −4 −8 −4 σ(8) z 00 −8 −16 −8 σ(9) z 0010 σ(10) z 000 σ(11) z 00 σ(12) z 0⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ Fig. 3 Ising Hamiltonian system for factoring 143 5 Quantum algorithms for discrete logarithmic problems LetG=/angbracketleftg/angbracketrightbe a cyclic group of order pand gbe a generator of G. The discrete logarithmic problem (DLP) over Gis to find an integer rsuch that gr=yfor given y∈G. Diffie–Hellman key exchange protocol, ElGamal encryption and most elliptic curve cryptosystems are based on the difficulty of computing discrete logarithms. Atpresent, the best known classical algorithm for solving DLP is the so-called index- calculate method (ICM) that requires sub-exponential classical operations. In 1994, Shor [ 3] put forward a polynomial time quantum algorithm to solve the discrete logarithmic problem in group. Proos et al. [ 42] further extended Shor’s quan- tum DLP algorithm to elliptic curves [ 42–44]. In 2012, Myasnikov et al. proposed a quantum algorithm for the DLP over matrices of finite group rings [ 45].
Question 69multiple-choice
Quantum error correction uses mathematical structures to protect quantum information against noise, often leveraging symmetries from group theory. Advanced code families can surpass traditional limitations by adapting subsystem dimensions and exploiting entanglement properties.
Which statement correctly characterizes how infinite-dimensional quantum codes covariant with respect to Lie groups can evade the Eastin-Knill theorem?
1) They utilize classical error correction protocols to bypass symmetry restrictions.
2) They rely on randomized code constructions to approximate transversal gates for any unitary.
3) They use finite-dimensional qudit systems to achieve perfect error correction for continuous groups.
4) They encode logical information exclusively in product states to avoid group covariance constraints.
5) They exploit subsystem swapping to circumvent the need for group representations.
6) They allow continuous symmetry groups and perfect erasure correction by employing infinite-dimensional systems, making exact transversal gates possible.
7) They restrict code families only to discrete group symmetries without using infinite-dimensional subsystems.
✓ Correct Answer:
The correct answer is 6) They allow continuous symmetry groups and perfect erasure correction by employing infinite-dimensional systems, making exact transversal gates possible..
📚 Reference Text:
subsystem dimension (Sec. VII). Leveraging tools from representation theory, we show that the lower bound on ϵworst becomes ϵworst≥1 2nmax ilndiþO/C181 ndL/C19 ; ð3Þ where diis the dimension of the ith physical subsystem. We also find lower bounds for the local subsystem dimension that depend on the number of logical qubits and the code ’s infidelity. This result also applies to the case when eachgate can be approximated with a discrete sequence oftransversal operations to arbitrary accuracy, as in the context of the Solovay-Kitaev theorem. Furthermore, using randomized constructions, we prove the existence of Uðd LÞ-covariant code families which achieve arbitrarily small infidelity in the limit of large subsystem dimension. In addition, we exhibit a simpleUðd LÞ-covariant code family, whose code words are generalized Wstates, such that ϵworst approaches zero as the number of subsystems napproaches infinity. D. Covariant codes from multiqubit codes We also construct codes that are covariant with respect to any group Gadmitting a Haar measure, where both the logical system and the physical subsystems transform as the regular representation of G(see Sec. VIII). In this con- struction, the dimension of each subsystem is the order jGj of the group when Gis finite and infinite when Gis a Lie group. Using this formalism, we construct natural generaliza- tions of well-known families of qubit codes, such as thebit-flip and phase-flip codes, with the qubits replaced by jGj-dimensional systems. These codes admit transversal logical gates representing each element of G. When Gis a Lie group, the qubits are replaced by infinite-dimensional systems such as rotors or oscillators. These infinite-dimensional codes circumvent the Eastin- Knill theorem —they are covariant with respect to a continuous symmetry group, yet erasure of a subsystemis perfectly correctable. III. SETUP AND NOTATION A. Approximate quantum error correction We define a code as a completely positive, trace- preserving map Ewhich assigns to each logical state on some abstract logical system La corresponding state on a physical system Aconsisting of nsubsystems A¼A 1⊗A2⊗/C1/C1/C1⊗An(Fig. 3). Unless explicitly stated otherwise, we consider in this paper codes that are isometric, i.e., thatmap any pure logical state jxi Lto a pure physical state fjψxiAg. The latter is called a code word . The span of all the code words fjψxiAgforms the code space . While typically encountered codes are isometric, it is useful in some instances to consider more general encoding maps such as encodings that symmetrize a state with a referenceframe [12]. The noise channel is the process to which the physical system is exposed, which might cause the informationencoded in it to get degraded. It is a quantum channel N A→Bmapping the physical system to physical system B. (The system Bmight be the same as A, but it might be different; for instance, Bmight include a register which remembers which type of error occurred or which sub- system was lost.) To study the approximate error-correction properties of a code, we need to quantify the approximation quality using distance measures between states and channels. Proximitybetween quantum states can be quantified
Question 70multiple-choice
Matrix Lie groups are mathematical structures that combine group theory and differential geometry, appearing frequently in physics and engineering to model symmetries and continuous transformations. Compactness is a key property in the representation theory of Lie groups, ensuring mathematical tractability and physical relevance.
Which property distinguishes compact matrix Lie groups from general matrix Lie groups, making their representation theory particularly "well behaved"?
1) Associativity of multiplication
2) Existence of an identity element
3) Being defined as a subgroup of GL(d, C) or GL(d, R)
4) Smooth manifold structure
5) Closure under group operations
6) Being both closed and bounded in the space of d×d matrices
7) Allowing infinite-dimensional representations
✓ Correct Answer:
The correct answer is 6) Being both closed and bounded in the space of d×d matrices.
📚 Reference Text:
subgroupHof a groupGis a subset of Gwhich is also a group. Definition 3. Amatrix Lie group Gis a closed subgroup ofGL(d;C)(orGL(d;R)), where by closed we mean that ifAm2Gis a sequence of matrices with limm!1Am= A2GL(d;C), thenA2G. Matrix Lie groups have the key feature that they also formsmooth manifolds , or hypersurfaces. In practice, this means that we can parameterize matrix Lie groups using coordinates, and we can take derivatives along paths in the group just as one can compute tangent vectors along paths embedded in surfaces. The smooth manifold structure turns the potentially unwieldy problem of understanding groups with uncountably many elements into a tractable one: we get to use powerful tools from not just algebra, but analysis, geometry, and topology as well. It is worth mentioningthatingreatergenerality, aLiegroupisagroup Gthat is also a manifold, where the group multiplication and inversion are continuous operations. Indeed, there are Lie groups that are not matrix Lie groups, but most of the physically relevant examples like the unitary groups U(n), orthogonal groups O(n), or special linear groups SL(n) form matrix Lie groups, so we restrict our attention to this more concrete class.3We refer the reader to Box 14 at the end of this work where we present some important exam- ples of matrix Lie groups and their key properties. Now, finite discrete groups are “well behaved” in com- parison to their infinite discrete cousins, in the sense that we can say many things about their representation the- ory: for instance, we know their representations are com- pletely reducible (Theorem 4), can always be converted into unitary representations (Theorem 5), and we know how many irreducible representations they have (Defini- tion 13). While we do not define these terms now, they will be presented below, so do not worry if these do not make sense now. The analogous condition to yield “well behaved” matrix Lie groups is a sort of topological proxy for finiteness: compactness . Compact matrix Lie groups will have the nicest representation theory, and luckily for quantum researchers, unitary groups are compact. Below we present a special version of the definition of compact- ness given by the Heine-Borel theorem. 3In fact, via a corollary of the Peter-Weyl theorem [59], every com- pact Lie group is a matrix Lie group. 9 Definition 4. A matrix Lie group Gis called compact if it is a closed and bounded subset of the vector space of the ddmatricesMd(C)(orMd(R)). It is useful to note that any putative Lie groups defined byequationsandcontinuousoperations(likemultiplication or taking adjoints) can be readily shown to be closed: for instance, if we consider a sequence Umof unitary matri- ces converging to a matrix U, then since taking adjoints U7!Uyand multiplication U7!UyUare continuous op- erations, we can take the limit of the equation Uy mUm=1 asm!1to see that UyU=1, meaning the unitary group is closed. Boundedness can be readily checked from standard linear algebra knowledge: for instance, since the operator normkUk= 1, the unitary group is bounded. Thus, the unitary group U(d)(and consequently the spe- cial
Question 71multiple-choice
In quantum computing, implementing gates such as the Quantum Fourier Transform (QFT) often involves combining controlled rotations and free evolution in pulse sequences. Optimizing these sequences is critical for achieving high-fidelity gate operations within minimal time.
When synthesizing a QFT gate using radio frequency (rf) pulse sequences, which parameter choice ensures minimum total gate duration while maintaining positive evolution times for different global phase options?
1) Selecting the smallest possible rotation angles regardless of evolution time
2) Maximizing the amplitude of the rf pulses without optimizing time parameters
3) Using only single rf pulses and avoiding composite pulse techniques
4) Choosing global phases that result in negative evolution times
5) Ignoring direction cosines and optimizing only pulse amplitude
6) Setting all evolution times equal to zero for fast operation
7) Selecting solutions with positive evolution times t₁ and t₂ that minimize the sum Tₘ = t₁ + t₂ for each global phase
✓ Correct Answer:
The correct answer is 7) Selecting solutions with positive evolution times t₁ and t₂ that minimize the sum Tₘ = t₁ + t₂ for each global phase.
📚 Reference Text:
y y x x r . (20) where { } exp( ) iI is the operator of non selective rotations by angle around the axis and exp( )t qit H is the free evolution for time t. In the center of sequence (20), there is the rotation by the angle 22 around the axis with the direction cosines / and / along the axes x and z, respectively. The nonselective rotation s can be obtained using a simple or composite pulse of the rf field with a large amplitude [25]. Thus, to implemen t the QFT gate, it remains for us to determine the parameters of pulse sequence (20) by equating the sum of matrices (18) to (9). As a result , we obtain the system of equations 7 2216 12 1 12 2 2212 12 22112 1 2 1 2 2 22113 1 2 1 2 2 1 12 23cos 1 3cos 1 sin cos sin cos sin sin sin cos sin cos sin sin 3cos 1 3cos 1 sin cos sin cos sin cos sin cmmtt K E it t tt it t t t t t it t it t 2216 12 os 3cos 1 3cos 1 tt (21) The joint solution of these equations yields the desire d values of the parameter s (Table I). For each value of the global phase { / 6, 5 / 6, 9 / 6} , we select the solution s with positive evolution times 1t and 2t that yield the minimum sum m 1 2T t t and one solution with time mT next in magnitude but with the same global phase . [If we neglect the rf pulse length in (20), the n the total sequence duration is m 1 2T t t ; this value is the minimum tim e for implementation of the QFT gate with the use of the method under consideration . Here, w e use the notation mT instead of cT , because this value is only a rough estimate of critical time cT determined in Section III]. TABLE I. Parameter for implementing eff mH (9) with the use of pulse sequenc e (20) m 1 2 3,,m m m 1t 2t mT /6 1, 0, 0 1, 1, -1 /2 /2 -0.905 -0.984 0.790 4.764 0.105 3.822 3.441 0.503 4.267 7.548 7.71 8.05 5
Question 72multiple-choice
Quantum Fourier transforms (QFTs) play a central role in quantum algorithms for group-theoretic problems, with efficient circuit implementations crucial for practical applications. For certain non-abelian groups, such as the Heisenberg group Hp of order p³, specialized techniques are required to achieve efficient QFTs.
Which of the following statements correctly describes a feature of the efficient quantum Fourier transform circuit construction for the Heisenberg group Hp?
1) The circuit leverages a subgroup tower approach, starting with the abelian normal subgroup N∞ ≅ Zp × Zp, and extends to the full group using a transversal and representation decomposition.
2) The circuit directly performs the QFT on the entire group without decomposing into subgroups or using representation theory.
3) The efficient circuit requires exponentially many quantum gates in p to implement the QFT for Hp.
4) The subgroup used in the construction is non-abelian and does not have a direct product structure.
5) The construction does not utilize binary string encoding for group elements in the quantum registers.
6) The QFT for Hp is performed by measurement-based quantum computation rather than unitary gates.
7) The representation decomposition is not needed for efficient implementation in non-abelian groups.
✓ Correct Answer:
The correct answer is 1) The circuit leverages a subgroup tower approach, starting with the abelian normal subgroup N∞ ≅ Zp × Zp, and extends to the full group using a transversal and representation decomposition..
📚 Reference Text:
now weare doing ab elian Fourier sampling over anon-abelian group, wehavetoconsider theeffectofapplying Ftoacosetof tAi,j,wheret= (0,0,τ)andτ∈Fp. Note thattAi,j={(µ,µi,/parenleftbigµ 2/parenrightbig i+µj+τ) :µ∈Fp}. Weobtain F|tAi,j/an}bracketri}ht=1 p2/summationdisplay a,b,c∈Fpωcτ /summationdisplay µ∈Fpω(a+bi+cj−ci 2)µ+ciµ2 |a,b,c/an}bracketri}ht. Hence, the probability of observing a particular triple (a,b,c)isp−4|/summationtext µ∈Fpω(a+bi+cj−(ci)/2)µ+ciµ2|2. If c/ne}ationslash= 0, this is a quadratic Weil sum and we can use Fact 1 to conclude t hat the probability of observing (a,b,c)is given by p−3, independent of i,j. The probability of observing (a,b,c),c/ne}ationslash= 0is1−1 p. If c= 0, only terms of the form (−bi,b,0)show up. These terms do give information about i; however, the probability of observing such a term is1 p. Thus, the forgetful abelian method gives exponentially sm all information about i. 2.5 Efficient quantum circuits forthe FourierTransform on Hp The fact that any QFTfor any finite group is a unitary matrix (when properly normal ized) makes this class of transformations an important source of transforma tions a quantum computer can carry out. The problem of finding efficient implementations of QFTs interms of quantum circuits wasstudied previously, see [Høy97, Bea97, PRB99, HRTS03, MRR04]. From [MRR04, Theo rem 2] it follows that for any prime ptheQFTfor the Heisenberg group Hpcan by computed in polylog(p)operations. In the following we give an explicit description of an efficient quantum circuit which computes QFTHp. First, note that we are interestedinarealizationonaquantumcomputer whichwork sonqubits. Thismeansthatwehavetoembed thestates and transformations intoaregister of size 2nfor somepositive integer n. Inthefollowing wewill assumethatnisthesmallestintegersuchthat p<2nandwewillidentifythegroupelements (x,y,z)∈ Hp withasubset ofthebinarystrings oflength 3n: ineachofthethreecomponents wechoosethebasisvectors |0/an}bracketri}ht,...,|p−1/an}bracketri}htto represent the respective component of the element (x,y,z). The following proposition shows that a QFTforHpcan be implemented efficiently in termsof elementary quantu m gates. Proposition 1 Letpbeprime,let HpbetheHeisenberggroupoforder p3andletIrr(Hp) ={χa,b: (a,b)∈ F2 p} ∪ {ρk:k= 1,...,p−1}denote the irreducible representations of Hp. Then the QFTforHpwith respect to Irr(Hp)can be computed using O(log3p)elementary quantum gates. Proof: First we consider the normal subgroup N∞⊳Hpand compute a Fourier transform for this abelian group. Thisgroupisisomorphictoadirectproduct oftwocyc licgroups, i.e., N∞∼=Zp×Zp. Theelements ofN∞are given by N∞={(0,y,z) :y,z∈Zp}, i.e., we can identify the elements of N∞with those binary strings of length 3nwhich have trivial support on the first npositions. Note that the irreducible representations of N∞are given by ψa,bfora,b∈Zp, where ψa,b(0,y,z) := exp(2πi/p(ay+bz)) =ωay+bz p. SinceN∞isnormalthegroup Hpoperates ontheirreducible representations [CR62]. Weden otethisaction by “∗”, i.e., we have a map ∗:Hp×Irr(Hp)→Irr(Hp)which is explicitly given by (x,y,z)∗ψa,b= ψa,b−ax. Next, we choose as a transversal for N∞⊳Hpthe ordered list T= [(x,0,0) :x∈Zp]. We have to be able to efficiently implement the images of all irreduci ble representations of Hpevaluated at the 8 elements of T. This is required for the so-called ‘twiddle factors’ in the decomposition of QFTHpalong the subgroup tower {1}⊳N∞⊳Hp. Indeed, we construct a QFTadapted to this subgroup tower, see also [PRB99, MRR04]. We now use the following formula for imp lementing a QFTGwhich holds in the situation where wehave anabelian normal subgroup Nand an abelian factor group G/N: QFTHp=/parenleftbig 11|G/N|⊗QFTN/parenrightbig/parenleftBigg/circleplusdisplay t∈TΦ(t)/parenrightBigg/parenleftBig QFTG/N⊗11|N|/parenrightBig . HereΦdenotes an extension of the decomposition of the regular rep resentation of Ninto irreducibles. Denoting this direct sum by Λ, i.e.,Λ :=/circleplustext
Question 73multiple-choice
The Hidden Subgroup Problem (HSP) plays a central role in quantum computing, as it underpins several problems where quantum algorithms can outperform classical ones. Recent research has investigated the extension of HSP beyond groups to more general algebraic structures.
Which statement accurately reflects a significant finding regarding the generalization of the Hidden Subgroup Problem to universal algebras and the tractability of their 2-element powers?
1) There exist 2-element algebras for which quantum algorithms achieve super-polynomial speedup over classical algorithms.
2) All 2-element algebras are classically intractable, regardless of quantum tractability.
3) Quantum algorithms are known to efficiently solve the HSP for all non-abelian groups.
4) The generalization to universal algebras eliminates the distinction between classical and quantum tractability.
5) No classification of tractability has been established for 2-element algebras.
6) Super-polynomial speedup by quantum algorithms is only possible for algebras with more than two elements.
7) The HSP for abelian groups remains unsolved by both classical and quantum algorithms.
✓ Correct Answer:
The correct answer is 1) There exist 2-element algebras for which quantum algorithms achieve super-polynomial speedup over classical algorithms..
Quantum computing has revolutionized the study of computational complexity, especially for problems involving hidden subgroup structures in Abelian groups. Understanding the quantum query complexity for these problems is essential for evaluating the fundamental limits of quantum algorithms.
Which statement accurately describes a key theoretical advancement in the quantum query complexity of Simon’s Problem and its generalization to Abelian groups?
1) It establishes that classical algorithms can match quantum algorithms for all hidden subgroup problems.
2) It proves that no lower bound exists for quantum query complexity in hidden subgroup problems.
3) It shows that the quantum query complexity for non-Abelian groups is constant.
4) It demonstrates that the quantum query complexity is always exponential for any group structure.
5) It provides the first nontrivial lower bound on quantum query complexity for Simon’s Problem and an optimal lower bound (up to a constant factor) for any Abelian group.
6) It concludes that hidden subgroup problems cannot be solved by quantum algorithms faster than classical ones.
7) It determines that quantum query complexity is irrelevant for problems involving group structures.
✓ Correct Answer:
The correct answer is 5) It provides the first nontrivial lower bound on quantum query complexity for Simon’s Problem and an optimal lower bound (up to a constant factor) for any Abelian group..
📚 Reference Text:
Title: A Quantum Lower Bound for the Query Complexity of Simon's Problem Year: 2005 Paper ID: 6e55023da9882955880cdf4e20260d1f7310c2a2 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/6e55023da9882955880cdf4e20260d1f7310c2a2 Abstract: Simon in his FOCS’94 paper was the first to show an exponential gap between classical and quantum computation. The problem he dealt with is now part of a well-studied class of problems, the hidden subgroup problems. We study Simon’s problem from the point of view of quantum query complexity and give here a first nontrivial lower bound on the query complexity of a hidden subgroup problem, namely Simon’s problem. More generally, we give a lower bound which is optimal up to a constant factor for any Abelian group.
Question 75multiple-choice
Complexity theory examines the relationships between various computational problems and the classes that describe their difficulty. Understanding the placement of problems like Graph Isomorphism and group-theoretic challenges within these classes is fundamental to assessing their computational boundaries.
Which of the following statements is true regarding the class SPP and its relationship with other complexity classes and lowness properties?
1) SPP contains all languages that are low for NP and is strictly larger than LWPP.
2) SPP is defined using gap-definable functions where the gap must be 0 for non-members and depends on an arbitrary function for members.
3) Every language in SPP is low for GapP, meaning adding it as an oracle does not increase the power of GapP.
4) SPP is equivalent to UP, and all problems in SPP have at most one accepting computation path.
5) Relativizing SPP to oracles destroys its containment relationships with UP and LWPP.
6) SPP is the same as PP, as both are characterized by gap-definable functions with positive gaps.
7) SPP only contains languages that are recognized by deterministic polynomial-time algorithms.
✓ Correct Answer:
The correct answer is 3) Every language in SPP is low for GapP, meaning adding it as an oracle does not increase the power of GapP..
📚 Reference Text:
question in [13] (also see [7]). As a consequence it follows that GI is in and low for ⊕P (in fact, GI ∈Mod kP for eachk≥2), C =P etc. Previously, only a special case of Graph Isomorphism, namely Tournament Isomorphism, was known to be in ⊕P.1 What we prove is a more general result: we show that a generic problem FIND-GROUP is in FPSPPas a consequence of which GI and some other algo- rithmic problems on permutation groups that are not known to have polynomial- time algorithms also turn out to be in SPP. In particular, as another corollary, we show that the hidden subgroup problem ( HSP) over permutation groups is in FPSPP. The hidden subgroup problem is of interest in the area of quantum algorithms. 2 Preliminaries and Notation Let Σ = {0,1}be the finite alphabet. Let log denote logarithm to base 2. Let FP denote the class of polynomial-time computable functions and NP denotes all languages accepted by polynomial-time nondeterministic Turing machines. LetZdenotes the set of integers. A function f: Σ∗→Zis said to be gap- definable if there is an NP machine Msuch that, for each x∈Σ∗,f(x) is the difference between the number of accepting paths and the number of rejecting paths ofMon inputx. Let GapP denote the class of gap-definable functions [7]. For each NP machine MletgapMdenote the GapP function defined by it. The language class PP is defined as follows: Lis in PP if there is an f∈GapP such thatx∈Lif and only if f(x)>0. The language classes UP, SPP and LWPP are defined using GapP func- tions [7].Lis in UP if there is an NP machine MacceptingLsuch thatM has at most one accepting path on any input. Lis in SPP if there is an NP machineMsuch thatx∈Limplies that gapM(x) = 1, andx/negationslash∈Limplies that 1Tournament Isomorphism in ⊕P follows because any tournament has an odd number of automorphisms. There are special cases of Graph Isomorphism, e.g. Graph Isomorphism for bounded-degree graphs or bounded genus graphs, that have polynomial-time algorithms. 2 gapM(x) = 0.Lis in LWPP if there are an NP machine Mandh∈FP such that x∈Limplies that gapM(x) =h(0|x|), andx/negationslash∈Limplies that gapM(x) = 0. For L∈SPP (or in LWPP) we say that the language Lisaccepted by the machine M. The containments UP ⊆SPP⊆LWPP is shown in [7]. We say that fis in GapPA, for oracle A⊆Σ∗, if there is an NPAmachine MAsuch that, for each x∈Σ∗,f(x) is the difference between the number of accepting paths and the number of rejecting paths of MAon inputx. For any oracle A, we can define the standard relativized classes UPA, SPPA, and LWPPAand we can easily see the containments UPA⊆SPPA⊆LWPPAfor any oracleA. We say that A⊆Σ∗islowfor PP if PPA= PP. In [7] it is shown that every language in LWPP is low for PP. Similarly, we say that A⊆Σ∗islowfor GapP if GapPA= GapP. Again, it is shown in [7] that Ais low for GapP if and only if A∈SPP. LetMbe an oracle NP machine, let A∈NP be accepted by an NP
Question 76multiple-choice
Lattice gauge theories are important tools for simulating non-perturbative phenomena in high-energy and condensed matter physics. Quantum computing offers new avenues for implementing these models using discrete quantum systems like qubits.
Which of the following statements best describes a key advantage of recasting gauge field theories as quantum link models in Minkowski space for quantum simulation?
1) It allows the use of bosonic operators exclusively for simulation.
2) It eliminates the need for gauge invariance in digital simulations.
3) It restricts simulations to classical computers rather than quantum devices.
4) It enables only static, equilibrium calculations without real-time dynamics.
5) It requires continuous variables for representing the gauge fields.
6) It makes real-time evolution feasible and compatible with quantum qubit algorithms.
7) It prevents the implementation of Suzuki-Trotter expansions in qubit circuits.
✓ Correct Answer:
The correct answer is 6) It makes real-time evolution feasible and compatible with quantum qubit algorithms..
📚 Reference Text:
Title: Lattice Gauge Theory for a Quantum Computer Year: 2020 Paper ID: 1d5e75bcbbee79fc5e2c062612dc903b49b7bbfe Source: semantic-scholar URL: https://www.semanticscholar.org/paper/1d5e75bcbbee79fc5e2c062612dc903b49b7bbfe Abstract: The quantum link~\cite{Brower:1997ha} Hamiltonian was introduced two decades ago as an alternative to Wilson's Euclidean lattice QCD with gauge fields represented by bi-linear fermion/anti-fermion operators. When generalized this new microscopic representation of lattice field theories is referred as {\tt D-theory}~\cite{Brower:2003vy}. Recast as a Hamiltonian in Minkowski space for real time evolution, D-theory leads naturally to quantum Qubit algorithms. Here to explore digital quantum computing for gauge theories, the simplest example of U(1) compact QED on triangular lattice is defined and gauge invariant kernels for the Suzuki-Trotter expansions are expressed as Qubit circuits capable of being tested on the IBM-Q and other existing Noisy Intermediate Scale Quantum (NISQ) hardware. This is a modest step in exploring the quantum complexity of D-theory to guide future applications to high energy physics and condensed matter quantum field theories.
Question 77multiple-choice
Quantum computing relies on maintaining quantum coherence, but real-world systems are often subject to decoherence from environmental interactions. Decoherence-free subspaces (DFS) and robust circuit designs are essential strategies for reliable quantum computation.
Which approach enables the implementation of the Quantum Fourier Transform (QFT) in quantum networks where multiple qubits experience collective decoherence, thus improving the reliability of quantum algorithms?
1) Utilizing classical error correction codes alongside quantum circuits
2) Employing physical isolation of individual qubits in separate environments
3) Applying active error correction with frequent syndrome measurements
4) Encoding quantum information within decoherence-free subspaces (DFS) tailored for collective decoherence
5) Running quantum algorithms at extremely low temperatures to minimize noise
6) Increasing the number of ancillary qubits to detect and correct all errors
7) Using quantum teleportation to bypass environmental interactions
✓ Correct Answer:
The correct answer is 4) Encoding quantum information within decoherence-free subspaces (DFS) tailored for collective decoherence.
📚 Reference Text:
Title: Quantum Fourier Transform in a Decoherence-Free Subspace Year: 2004 Paper ID: 019ffe375decaf8fd78e76fdfeef5fccff06de9a Source: semantic-scholar URL: https://www.semanticscholar.org/paper/019ffe375decaf8fd78e76fdfeef5fccff06de9a Abstract: Quantum Fourier transform is of primary importance in many quantum algorithms. In order to eliminate the destructive effects of decoherence induced by couplings between the quantum system and its environment, we propose a robust scheme for quantum Fourier transform over the intrinsic decoherence-free subspaces. The scheme is then applied to the circuit design of quantum Fourier transform over quantum networks under collective decoherence. The encoding efficiency and possible improvements are also discussed.
Question 78multiple-choice
Group symmetries play a fundamental role in quantum physics and quantum machine learning, where the invariance of labels under group actions ensures robust classification and meaningful physical properties. Both continuous and discrete groups underpin the classification of quantum states and operations on Hilbert spaces.
Which statement accurately describes the role of local unitaries in the classification of multipartite entanglement in quantum systems?
1) Labels for multipartite entanglement are invariant under local unitaries acting independently on each qubit, corresponding to the group U(2) ×.. × U(2).
2) Local unitaries always change the entanglement classification of a quantum state.
3) Multipartite entanglement labels are only invariant under global orthogonal transformations, not under local unitaries.
4) The invariance of labels under local unitaries is specific to classical systems, not quantum systems.
5) Local unitaries are irrelevant in determining the separability of quantum states.
6) Entanglement measures are only invariant under the symmetric group Sn.
7) Only ground states in the Heisenberg XXX model exhibit invariance under local unitaries, not multipartite entangled states.
✓ Correct Answer:
The correct answer is 1) Labels for multipartite entanglement are invariant under local unitaries acting independently on each qubit, corresponding to the group U(2) ×.. × U(2)..
📚 Reference Text:
the unitary group on H. •Representation: U:G7!U(d), wheregi=UgiUy g. Orthogonal transformations: Consider a problem of classifying orthogonal (real-valued) states from Haar- random states. The dataset here is composed of states with label yi= 0(yi6= 0) ifiis a real-valued state (a Haar random state). Here, the labels yi= 0are invariant under the action of any orthogonal unitary, as conjugated a real-valued state by a real-valued unitary yields a real-valued state. Note that here f(UiUy) =f(i), but in general UiUy6=i. •States:Statesi2D(H). •Labels:Orthogonal yi= 0and mixed yi6= 0. •Group:G=O(d), the orthogonal group on H •Representation: U:G7!O(d), wheregi=UgiUy g Local unitary transformations and the XXX model: Consider the problem of classifying ground states of the Heisenberg XXXmodelH=JPn j=1(XjXj+1+YjYj+1+ZjZj+1). Here,yi= 0(yi= 1) ifiis a ferromagnetic (antiferromagnetic) ground state of HwithJ < 0(J > 0). SinceHcommutes with the total magnetization operators Sx=Pn j=1Xj,Sy=Pn j=1Yj,Sz=Pn j=1Zj, then the labels are invariant under the action of the same local unitary acting on all qubits. That is, f((Nn iU)i(Nn iUy)) =f(i)for any local unitaryU. •States:Ground states of the XXXchaini2D(H) •Labels:Ferromagnetic yi= 0and antiferromagnetic yi= 1 •Group:G=U(2) •Representation: U:G7!U(d), wheregi= (Ug Ug)i(Ug Ug)y Local unitary transformations and multipartite entanglement: Consider the problem of classifying pure quantum states according to the amount of multipartite entanglement they posses. Here, yi= 1if the states posses a large amount of multipartite entanglement (according to some measure), while yi= 0if the states are separable. Since local unitaries do not change the multipartite entanglement in a quantum state, then we have that f((Nn jUj)i(Nn jUy j)) =f(i)for any local unitary Ujacting on the j-th qubit. •States:Pure states i2D(H) •Labels:yi2[0;1], where 0 means separable and 1 means “highly entangled” •Group:G=U(2)U(2), (ntimes) •Representation: U:G7!U(d), where (g1;:::;gn)i= (Ug1 Ugn)i(Ug1 Ugn)y 42 Example 13: The usual suspects: commonly appearing discrete groups While a veritable zoo of group symmetries can (and do!) appear in the wild, any QML practicioner should be familiar with some especially common species. Consider this a beginner’s field guide to some frequently appearing groups and a few classic applied locations they have been observed. Note that discrete group representations often appear in disguise as subgroups of continuous groups (which is inevitable, because as we described earlier, we care about unitary representations on Hilbert spaces). For example, a =2pulse instantiates a rotation by =2on the Bloch sphere for a single qubit—the possible actions of strings of =2pulses generate the group of rotations Z4. Discrete groups •Zn, the cyclic group of integers modulo n. –Type, size, # irreps: Abelian,jZnj=n, n irreps (given by roots of unity) –Where you might find them: translations on periodic lattices, rotations of 2D regular polygons, roots of unity, parity transformations –Useful fact: Every finite abelian group is a direct product of cyclic groups, so finite abelian sym- metries can often be studied by restricting to cyclic groups. •Sn, the symmetric group on nletters (aka the group of all permutations on a set of size n). –Type, size, # irreps: nonabelian for n>3,jSnj=n!, integer partitions ofn ∗A tuple of positive integers
Question 79multiple-choice
Finite W-algebras are algebraic structures that arise from Lie algebras through reductions and play a significant role in representation theory and quantum field theory. Modern approaches to their study involve cohomological and geometric methods, including BRST quantization and Poisson geometry.
Which of the following statements best describes the role of the BRST differential in the quantization of finite W-algebras?
1) It splits into two commuting differentials, forming a double complex whose cohomology can be computed using spectral sequences.
2) It acts solely as a grading operator on the universal enveloping algebra without contributing to cohomology.
3) It defines a filtration that trivializes the algebraic structure, making spectral sequences unnecessary.
4) It generates finite dimensional irreducible representations directly, bypassing cohomological methods.
5) It identifies the Kirillov Poisson structure with the group manifold itself, removing the need for reduction.
6) It replaces the need for Hamiltonian reduction through explicit Fock space constructions.
7) It only applies to infinite W-algebras and cannot be used for finite cases.
✓ Correct Answer:
The correct answer is 1) It splits into two commuting differentials, forming a double complex whose cohomology can be computed using spectral sequences..
📚 Reference Text:
systems are r eductions of a system describing a free particle moving on a group manifold. This a llows us to give general formulas for the solution space of such systems. In the second part of the paper we BRST quantize the finite Walgebras. The nontrivial part of this is of course calculating the BRST coh omology and its algebraic structure. Since the BRST differential is a sum of two other di fferentials one can associate a double complex to the BRST complex. In order to ca lculate the BRST cohomology one can then use the theory of spectral sequences . There is a choice to be made between one out of two spectral sequences that one c an associate to a double complex. These spectral sequences must give the same final answer for the BRST cohomology, as is well known from the theory of spectral sequences, but for the calculation it is crucial which one one takes. The choice we m ake is different from the one made by Feigin and Frenkel and allows us to quantize any fin iteWalgebra and reconstruct its algebraic structure. In order to make our co nstruction more explicit we calculate all finite quantum Walgebras that can be obtained from sl2,sl3andsl4. In the third and last part of the paper we discuss the represen tation theory of finite Walgebras. Crucial for this is a quantum version of the genera lized Miura transfor- mation which embeds any finite Walgebra into the universal enveloping algebra of some (semi)simple Lie algebra. An arbitrary representatio n of this Lie algebra there- fore immediately yields a representation of the finite Walgebra. This also allows us 4 to derive Fock realizations for arbitrary finite Walgebras since Fock realizations for simple Lie algebras are well known. This replaces the cumber some construction of W algebras as commutants of screening operators. As an illust rative example we realize the finite dimensional representations of the finite Walgebra ¯W(2) 3as a subrepresen- tation of certain Fock realizations. In principle this prov ides the first term of a Fock space resolution of these representations [14]. We have tried to keep the paper as self-contained as possible and give full proofs of the main assertions. 2. Basics of Kirillov Poisson structures and Poisson reduct ion In order to make the paper reasonably selfcontained we briefl y discuss in this sec- tion the Kirillov Poisson structure on a Lie algebra and Pois son reduction of Poisson manifolds. These two concepts will then be put together in th e remainder of the paper. 2.1. Kirillov Poisson structures Let (M,{.,.}) be a Poisson manifold, that is {.,.}is a Poisson bracket on the space C∞(M) ofC∞functions on M, andGa Lie group. Also let Φ : G×M→Mbe a smooth and proper action of GonMwhich preserves the Poisson structure, i.e. Φ∗ g{φ,ψ}={Φ∗ g(φ),Φ∗ g(ψ)} (2.1) where Φ∗ gis the pullback of Φ g:M→M. Physically this implies that if γ(t) is a solution of the equations of motion w.r.t. some G-invariant Hamiltonian H
Question 80multiple-choice
Quantum topology investigates properties of 3-manifolds using quantum-inspired mathematical frameworks, often leveraging fusion categories to define computable invariants. Computational challenges in this field arise from both the quantum algebraic input and the underlying topological complexity of manifolds.
Which statement accurately describes the computational complexity and parameterization of algorithms for computing state sum invariants from Tambara-Yamagami categories on triangulated 3-manifolds?
1) The computation is #P-hard in general, but becomes fixed parameter tractable when parameterized by the first Betti number with Z/2Z coefficients.
2) The computation is always polynomial-time regardless of the manifold's topological properties.
3) The computational hardness originates solely from the quantum algebraic structure of the fusion categories.
4) There are currently no known algorithms that achieve fixed parameter tractability for this problem.
5) The complexity depends exponentially on the width of the triangulation rather than any topological parameter.
6) The first Betti number cannot be computed efficiently from combinatorial data of the manifold.
7) State sum invariants from Tambara-Yamagami categories are only defined for manifolds with trivial first homology.
✓ Correct Answer:
The correct answer is 1) The computation is #P-hard in general, but becomes fixed parameter tractable when parameterized by the first Betti number with Z/2Z coefficients..
📚 Reference Text:
Title: An algorithm for Tambara-Yamagami quantum invariants of 3-manifolds, parameterized by the first Betti number Year: 2023 Paper ID: 937b79eace34a89a7b58a9274ca096b6d8316f24 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/937b79eace34a89a7b58a9274ca096b6d8316f24 Abstract: Quantum topology provides various frameworks for defining and computing invariants of manifolds inspired by quantum theory. One such framework of substantial interest in both mathematics and physics is the Turaev-Viro-Barrett-Westbury state sum construction, which uses the data of a spherical fusion category to define topological invariants of triangulated 3-manifolds via tensor network contractions. In this work we analyze the computational complexity of state sum invariants of 3-manifolds derived from Tambara-Yamagami categories. While these categories are the simplest source of state sum invariants beyond finite abelian groups (whose invariants can be computed in polynomial time) their computational complexities are yet to be fully understood. We first establish that the invariants arising from even the smallest Tambara-Yamagami categories are #P-hard to compute, so that one expects the same to be true of the whole family. Our main result is then the existence of a fixed parameter tractable algorithm to compute these 3-manifold invariants, where the parameter is the first Betti number of the 3-manifold with Z/2Z coefficients. Contrary to other domains of computational topology, such as graphs on surfaces, very few hard problems in 3-manifold topology are known to admit FPT algorithms with a topological parameter. However, such algorithms are of particular interest as their complexity depends only polynomially on the combinatorial representation of the input, regardless of size or combinatorial width. Additionally, in the case of Betti numbers, the parameter itself is computable in polynomial time. Thus while one generally expects quantum invariants to be hard to compute classically, our results suggest that the hardness of computing state sum invariants from Tambara-Yamagami categories arises from classical 3-manifold topology rather than the quantum nature of the algebraic input.
Question 81multiple-choice
In quantum computing, problems like the Dihedral Coset Problem (DCP) involve extracting hidden parameters from quantum states, and the design of quantum algorithms is constrained by foundational principles such as unitarity and the no-cloning theorem. Understanding these limitations is crucial for developing effective quantum algorithms.
Which statement most accurately describes a fundamental limitation encountered when attempting to produce additional DCP samples or extract the hidden parameter a from existing quantum samples?
1) Using classical copying techniques, one can reliably duplicate DCP samples for any unknown parameter a.
2) The no-cloning theorem applies only to entangled quantum states and does not affect DCP samples.
3) Unitary quantum operations can always extract the parameter a into a register from any list of DCP samples.
4) If the parity of a is known, then any DCP sample can be cloned without restriction.
5) The linearity of quantum mechanics allows unlimited copying of quantum information encoded in coset states.
6) Quantum measurements are sufficient to deterministically compute a from a single DCP sample.
7) It is impossible to use unitary quantum operations to clone unknown DCP samples or extract a into a register without violating quantum mechanical principles such as linearity and the no-cloning theorem.
✓ Correct Answer:
The correct answer is 7) It is impossible to use unitary quantum operations to clone unknown DCP samples or extract a into a register without violating quantum mechanical principles such as linearity and the no-cloning theorem..
📚 Reference Text:
a α α(6.7) The sum of the observed exponents of the two registers gives a. Ifais known, then DCP samples for a, (∣ ⟩ ∣ ⟩)=+−ψx y x1 2,aααa α ;The dihedral hidden subgroup problem 15 are chosen from a set of mutually orthogonal states depending on the parameter a. By Proposition 6.1 , for each sample of the form (6.5), we can copy it to produce a sample of the form (6.6). □ Remark 6.8. Copying a DCP sample up to parity would allow us to determine the parity of a, and vice versa . Theorem 6.9. If a is unknown, there is no unitary operation, which from a list of DCP samples for a, copies an additional DCP sample for the same a, while leaving the list of DCP samples alone . Proof. Suppose that there is a unitary operator Uthat transforms ∣⟩ ∣⟩∣ ⟩ ∣⟩ ∣ ⟩ ∣() ⟩ ∣⟩∣ ⟩ ∣⟩ ∣⟩⋯= ⋯U A ψψ ψ ψ ψψ ψ ψ 0Σ ,aam a aaaam aa11(6.10) where (∣ ⟩ ∣ ⟩) == +−ψψ x y xaa ααa α ;1 2is a DCP sample for afixed, andαrandomly chosen for each such state. We are supposing that Uperforms the aforementioned operation for any(unknown) a. Thus, we also have that ∣ ⟩ ∣⟩∣ ⟩ ∣⟩ ∣ ⟩∣() ⟩ ∣⟩∣ ⟩ ∣⟩ ∣⟩⋯= ⋯U A ψψ ψ ψ ψψ ψ ψ 0Σ ,bbm b bbbbm bb11(6.11) for any other b. Taking the inner product of both sides of (6.10 ) and (6.11), we deduce ⟨∣⟩⟨∣⟩ ⟨∣⟩⟨∣⟩⟨∣⟩ ⟨∣⟩ ⟨() ∣() ⟩⋯= ⋯ψ ψ ψψ ψ ψ ψ ψ ψψ ψ ψ ψ ψ ΣΣ .ab am bm ab ab am bm ab aabb11 11 2(6.12) However, there are choices of ψψ,ai bifor=im 1,…, , andψψ,ab, which do not satisfy (6.12 ) from (5.5). We may thus suppose without loss of generality that ⟨∣⟩≠ψψ 0,1ai bifor all=iN 1,…, , and hence, (6.12 ) becomes ⟨∣⟩⟨∣⟩ ⟨() ∣() ⟩=ψψ ψψ ψ ψ ΣΣ .ab ab aabb2 We obtain a contradiction again by choosing ψaandψbso that⟨∣⟩≠ψψ 0,1ab as then ∣⟨ ∣ ⟩∣=ψψ1 2,ab(6.13) ∣⟨ ∣ ⟩ ⟨ ( )∣ ( )⟩∣ ≤ ψψ ψ ψ ΣΣ1 4.ab aabb2 (6.14) □ The following is another proof of Theorem 5.1 using the connection with quantum cloning. Theorem 6.15. There is no unitary operation to compute the value of a into a register from a list of DCP samples for a . Proof. Suppose that there is a unitary operator U, which has the e ffect ∣⟩ ∣ ⟩∣ ⟩ ∣⟩ ∣ () ⟩ ∣⟩⋯=UA ψ ψ ψ a 0Σ ,aam aa1(6.16) i.e.,Utakes a list of DCP samples for fixed but unknown a, a blank initialization state ∣⟩0, and an ancilla state ∣⟩A, and then computes ainto the blank register. Using an additional blank register and copying ∣⟩a, there is a unitary operator ′Uwith the e ffect: ∣⟩ ∣ ⟩ ∣ ⟩ ∣⟩ ∣⟩ ∣ ( ) ∣⟩ ∣⟩′⋯ =UAψ ψ ψ aa 00 Σ .aam aa1(6.17)
Question 82multiple-choice
Quantum complexity theory investigates the boundaries between quantum and classical computational models, particularly through problems that highlight computational separations under certain promises. The k-fold Forrelation problem and Quantum Circulant Optimization are significant in illustrating these separations and their implications for complexity classes.
Which statement accurately describes the consequence if a classical polynomial-time algorithm can solve Quantum Circulant Optimization efficiently under standard assumptions?
1) It proves that QNC is strictly harder than PromiseBQP.
2) It implies that PromiseBQP equals PromiseBPP, collapsing the quantum-classical separation for promise problems.
3) It shows that Forrelation is NP-complete.
4) It establishes that all quantum algorithms can be simulated by constant-depth classical circuits.
5) It demonstrates that quantum circuits are unnecessary for solving linear systems efficiently.
6) It means that PromiseQNC is disjoint from PromiseBPP.
7) It ensures that variational quantum algorithms outperform classical algorithms for all optimization problems.
✓ Correct Answer:
The correct answer is 2) It implies that PromiseBQP equals PromiseBPP, collapsing the quantum-classical separation for promise problems..
📚 Reference Text:
By reverse triangular inequality, we can further obtain |ˆα1| − |Φf1,f2,···,fk| 2≤ |ˆα1−Φf1,f2,···,fk|2≤ζ≤1 16⇒ |ˆα1| − |Φf1,f2,···,fk| ≤p ζ≤1 4. (C8) Hence, we see that the value of |ˆα1|is 1/4-close to |Φf1,f2,···,fk|. Given the promise that |Φf1,f2,···,fk|is either ≥3 5or ≤1 100, if|ˆα| ≥7 20, then |Φf1,f2,···,fk| ≥3 5; If|ˆα| ≤13 50, then |Φf1,f2,···,fk| ≤1 100. We see that k-foldForrelation can be reduced to QuantumCirculantOptim . If there exists a relativizing classical algorithm that can solve QuantumCirculantOptim in poly( κC, n) time, the same algorithm can solve k-foldForrelation equally efficiently, implying that PromiseBQP =PromiseBPP . Lastly, we note that the above proof works under the assumption that oracles and poly ndepth circuits are allowed in the preparation of UbinQuantumCirculantOptim . If only poly log ndepth is allowed for the preparation for Ub, only under the case where k∈poly log nisk-foldForrelation reducible to QuantumCirculantOptim . In this case, and disregarding the implementation depth of oracles, k-foldForrelation isQNC-complete, and we obtain the result that QuantumCirculantOptim isPromiseQNC -hard. Similarly, if there exists a classical algorithm that can solve QuantumCirculantOptim efficiently, the same algorithm can solve Forrelation equally efficiently, which implies that PromiseQNC ⊆PromiseBPP . [1] A. W. Harrow, A. Hassidim, and S. Lloyd, Quantum algorithm for linear systems of equations, Phys. Rev. Lett. 103, 150502 (2009). [2] M. Cerezo, A. Arrasmith, R. Babbush, S. C. Benjamin, S. Endo, K. Fujii, J. R. McClean, K. Mitarai, X. Yuan, L. Cincio, and P. J. Coles, Variational quantum algorithms, Nature Reviews Physics 3, 625 (2021). [3] X. Xu, J. Sun, S. Endo, Y. Li, S. C. Benjamin, and X. Yuan, Variational algorithms for linear algebra, Science Bulletin 66, 2181 (2021). [4] C. Bravo-Prieto, R. LaRose, M. Cerezo, Y. Subasi, L. Cincio, and P. J. Coles, Variational quantum linear solver (2020), arXiv:1909.05820 [quant-ph]. [5] J. Preskill, Quantum computing in the NISQ era and beyond, Quantum 2, 79 (2018). [6] J. R. McClean, S. Boixo, V. N. Smelyanskiy, R. Babbush, and H. Neven, Barren plateaus in quantum neural network training landscapes, Nature Communications 9(2018). [7] H.-Y. Huang, K. Bharti, and P. Rebentrost, Near-term quantum algorithms for linear systems of equations with regression loss functions, New Journal of Physics 23, 113021 (2021). [8] E. Tang, A quantum-inspired classical algorithm for recommendation systems, in Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing , STOC 2019 (Association for Computing Machinery, New York, NY, USA, 2019) p. 217–228. [9] N.-H. Chia, H.-H. Lin, and C. Wang, Quantum-inspired sublinear classical algorithms for solving low-rank linear systems (2018), arXiv:1811.04852 [cs.DS]. [10] A. Gily´ en, S. Lloyd, and E. Tang, Quantum-inspired low-rank stochastic regression with logarithmic dependence on the dimension (2018), arXiv:1811.04909 [cs.DS]. [11] A. Gily´ en, Z. Song, and E. Tang, An improved quantum-inspired algorithm for linear regression, Quantum 6, 754 (2022). [12] C. Shao and H. Xiang, Quantum circulant preconditioner for a linear system of equations, Phys. Rev. A 98, 062321 (2018). [13] Q. Zuo and T. Li, Fast and practical quantum-inspired classical algorithms for solving linear systems (2023),
Question 83multiple-choice
In group theory and quantum mechanics, unitary and special unitary groups play a fundamental role in understanding symmetry transformations of complex vector spaces. The mathematical properties of these groups have significant implications for the types of representations and physical phenomena they can describe.
Which statement is true regarding the group SU(d) for d ≥ 2?
1) SU(d) is compact, connected, and simply connected.
2) SU(d) is non-compact, disconnected, and abelian.
3) SU(d) is compact, not connected, and not simply connected.
4) SU(d) is abelian and consists only of diagonal matrices.
5) SU(d) contains all orthogonal transformations of real vector spaces.
6) SU(d) is isomorphic to the cyclic group Z_d.
7) SU(d) is non-compact but simply connected.
✓ Correct Answer:
The correct answer is 1) SU(d) is compact, connected, and simply connected..
📚 Reference Text:
= (1;:::;k)is called a partition if1++k=nand 1>:::k>0. Partitions, and thus irreps of Sn, are labeled by Young diagrams. –Where you might find them: Symmetric and antisymmetric vectors, determinants, more general permutations on tensor indices, combinatorial identities –Useful fact: Schur-Weyl duality tells us that the tensor representations U nofU(d)on(Cd) ncan be decomposed into representations of U(d)andSn, whereU(d)acts on one CdandSnpermutes tensor indices. •Dn, the dihedral group of symmetries of the regular n-gon. –Type, size, # irreps: nonabelian for n>3,jDnj= 2n,(n+ 3)=2ifnodd, (n+ 6)=2ifneven –Where you might find them: Molecular symmetry, finite subgroups of O(2) –Useful fact: Dnis generated by rotations of angle 2=nand reflections, and so on the complex plane is commonly thought of as the group generated by multiplication by e2i=nand complex conjugation. •Zn=Z Z, the additive group of integer vectors (k1;:::;kn) –Type, size: abelian,jZnj=1 –Where you might find them: Fourier series, translations on infinite lattices –Useful fact: The cyclic group Z`can be thought of a quotient group of the group of integers Z, or more geometrically, Z`is the group of translations on a ring with `sites (“periodic boundary conditions”). Likewise, if we quotient every Zin the translation group Zn, we get tori Z`1Z`2 Z`n. 43 Example 14: The usual suspects: commonly appearing continuous groups The field guide continues with some commonly appearing continuous groups. These species hold privileged positions in physics as symmetries enjoyed by a variety of differential equations, and they are often detected at the level of their Lie algebra as “infinitesimal symmetries”. While certainly the unitary groups are most important for QML, we would not put it past these other groups to sneakily appear in a variety of tasks. Perhaps you will tell us where you have caught them! Continuous groups •GL(d;C)andSL(d;C), the (complex) general and special linear groups –Topological info: Not compact, connected, simply connected –Lie algebra: gl(d) =Md(C), theddcomplex matrices, and sl(d) =fX2gl(d) : Tr[X] = 0g, the tracelessddcomplex matrices –Where you might find them: Change of bases, so basically everywhere. –Useful fact: The complexified Lie algebras u(d) C=gl(d) C=gl(d)and su(d) C=sl(d) C= sl(d), which means their (complex) representation theory is the same, even though gl(d)6=u(d)and sl(d)6=su(d)(u(d);su(d)are real vector spaces but not complex vector spaces). •U(d)andSU(d), the unitary and special unitary groups –Topological info (for SU(d)):Compact, connected, simply connected ∗U(d)=SU(d)=U(1), the circle group. The isomorphism follows immediately from det :U(d)! U(1).U(d)is not simply connected. –Lie algebra: u(d) =fX2Md(C) :X= Xyg, theddskew-hermitian matrices, and su(d) = fX2u(d) : Tr[X] = 0g, the traceless ddskew-hermitian matrices. Note that u(d);su(d)are real vector spaces: e.g. if X2su(d), theniX62su(d). –Where you might find them: Literally everywhere in quantum. –Useful fact: Wigner’s theorem assures us that every symmetry transformation on physical states which preserves the Hermitian inner product is either a unitary or antiunitary transformation. This in large part motivates the focus on unitary representations from a physical standpoint. •O(d)andSO(d), the orthogonal and special orthogonal groups –Topological info (for SO(d)):Compact, connected, not simply connected ∗Note that the orthogonal group O(d)consists of two disconnected copies of SO(d): matrices fR:R2SO(d)g, and matricesfSR:R2SO(d)and det(S)
Question 84multiple-choice
Quantum algorithms such as Shor’s threaten classical cryptographic systems by making integer factorization much more efficient. The practical implementation of these algorithms depends heavily on the complexity of quantum hardware and the types of quantum gates required.
Which modification to Shor’s quantum factorization algorithm could simplify hardware requirements by allowing the use of only simple phase rotation gates with states 0 and π?
1) Replacing modular exponentiation with classical arithmetic operations
2) Using Grover’s search algorithm in place of the period-finding subroutine
3) Substituting Toffoli gates for all controlled-NOT gates
4) Applying the SWAP gate for state preparation
5) Implementing the classical discrete Fourier transform
6) Using phase estimation with random unitary gates
7) Replacing the quantum Fourier transform with the quantum Hadamard transform in certain cases
✓ Correct Answer:
The correct answer is 7) Replacing the quantum Fourier transform with the quantum Hadamard transform in certain cases.
📚 Reference Text:
Title: A simplification of the Shor quantum factorization algorithm employing a quantum Hadamard transform Year: 2018 Paper ID: 9ad5c7402dda2a01df870dd3036c70ce398a10b6 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/9ad5c7402dda2a01df870dd3036c70ce398a10b6 Abstract: The Shor quantum factorization algorithm allows the factorization or large integers in logarithmic squared time whereas classical algorithms require an exponential time increase with the bit length of the number to be factored. The hardware implementation of the Shor algorithm would thus allow the factorization of the very large integers employed by commercial encryption methods. We propose some modifications of the algorithm by employing some simplification to the stage employing the quantum Fourier transform. The quantum Hadamard transform may be used to replace the quantum Fourier transform in certain cases. This would reduce the hardware complexity of implementation since phase rotation gates with only two states of 0 and π would be required.
Question 85multiple-choice
In the study of topological quantum order in lattice models, qudits are often associated with the edges of a lattice, and the ground state properties are defined via commuting projector Hamiltonians. These systems exhibit robust quantum error correcting capabilities due to their global, non-local nature.
Which statement accurately describes the condition known as TQO-1 in topologically ordered, frustration-free Hamiltonians?
1) The ground state energy is minimized only globally, not locally.
2) The ground state manifold is unique for all choices of sublattice.
3) Projectors Pv and Pf do not commute on overlapping regions.
4) Local operators can distinguish all ground states on any sublattice.
5) All ground states have indistinguishable reduced density matrices on any small sublattice, so no local operator can differentiate between them.
6) The ground state projector acts only on the vertices of the lattice.
7) Topological degeneracy arises from boundary conditions rather than global constraints.
✓ Correct Answer:
The correct answer is 5) All ground states have indistinguishable reduced density matrices on any small sublattice, so no local operator can differentiate between them..
📚 Reference Text:
qudits are conventionally defined to live on the edges of Λinstead of the vertices; for simplicity we use the same convention here.2We therefore associate to each edge e∈E(Λ)a qudit He=Cd,and take the total Hilbert space to be H=/circlemultiplytext e∈E(Λ)He. We consider Hamiltonians of the form H=/summationdisplay v∈V(Λ)(1−Pv) +/summationdisplay f∈F(Λ)(1−Pf), (5) wherePvis a projector that acts non-trivially only on edges which meet the vertex v, andPf is a projector that acts non-trivially only on the boundary edges of the plaquette f. We further demand that the Pv’s and thePf’s mutually commute and that the Hamiltonian be frustration free, i.e. that the ground states of Hare stabilized by each Pvand eachPf: Vg.s.={|ψ/angbracketright∈H:Pv|ψ/angbracketright=|ψ/angbracketrightandPf|ψ/angbracketright=|ψ/angbracketright,∀v∈V(Λ),f∈F(Λ)}. (6) Denote the projection onto Vg.s.byP, which can be written as P=/productdisplay v∈V(Λ)Pv/productdisplay f∈F(Λ)Pf. (7) LetAbe a sublattice of Λof size/lscript×/lscript, denote by V(A)◦the subset of V(A)that are in the interior ofA(which is of size (/lscript−2)×(/lscript−2)), and define PA=/productdisplay v∈V(A)◦Pv/productdisplay f∈F(A)Pf. (8) We can now state the definition of TQO that we will use. Definition 2.2 (Topological Quantum Order [5]) .A Hamiltonian which is frustration-free is said to have topological quantum order (TQO) if there is a constant α > 0such that for any /lscript×/lscript sublatticeAwith/lscript≤Lα, the following hold. 2The choice of whether the qudits live on the edges or vertices of the lattice is arbitrary and makes no difference to the definition. Accepted in Quantum2020-09-17, click title to verify. Published under CC-BY 4.0. 4 •TQO-1: For any operator Oacting onA, POP =cOP, (9) wherecOis some complex number. •TQO-2: IfBis the smallest square lattice whose interior properly contains A,3then Tr¯A(P) andTr¯A(PB)have the same kernel, where ¯Ais the complement of AinΛ. TQO-1 heuristically corresponds to the statement that a sufficiently local operator cannot be used to distinguish between two orthogonal ground states because they differ only in their global, “topological” properties. Furthermore, ground state denegeracy is “topologically protected” in systems satisfying TQO-1 in the sense that perturbations by local operators can induce energy levelsplittingonlynon-perturbatively, oratsomelargeorderinperturbationtheorywhichincreases with the size of the lattice. It is straightforward to show that TQO-1 is equivalent to the condition that all normalized ground states |ψ/angbracketright∈Vg.s.have the same reduced density matrix on A. TQO-2 is the statement that the local ground state spaces and the global one should agree. We emphasize that TQO-2 can be violated at regions with non-trivial topologies, which is why one restricts to square lattices. Remark 2.3. For our purposes, TQO-1 and QECC are morally interchangeable. Indeed, if H is any Hamiltonian4withPthe projection onto the ground space Vg.s., then the following are equivalent. 1. The Hamiltonian Hhas TQO-1. 2. The Hamiltonian Hprovides a QECC with code subspace Vg.s.. There exists an α>0such that the code can correct any error ρ/mapsto→/summationtext iEiρE† ifor which every combination E† iEjis supported on an /lscript×/lscriptsublatticeAwith/lscript≤Lα. In §3.2, we will prove a theorem for Kitaev’s finite group models which simultaneously implies TQO-1 and TQO-2, and so by the above remark also implies that the model furnishes a QECC. 2.3 Kitaev’s finite group lattice model We now turn to Kitaev’s
Question 86multiple-choice
In quantum photonic computing, photon indistinguishability is central to achieving high-fidelity quantum operations, and errors in this property introduce specific challenges for modeling and simulation of quantum circuits. Specialized frameworks are used to map such errors to standard quantum error types for efficient analysis.
Which approach enables efficient simulation of large-scale quantum circuits with low-order photon distinguishability errors by combining stabilizer formalism with minimal internal state tracking per photon?
1) Using only dual-rail projection to eliminate all internal state information
2) Applying amplitude damping channels to model photon loss exclusively
3) Employing a full density matrix representation for every photon in the circuit
4) Probabilistically applying Pauli errors and a distinguishability operator, tracking stabilizers and minimal internal state per photon
5) Disregarding all distinguishability errors after initial state preparation
6) Using only phase-flip errors to represent all error types
7) Modeling errors as purely classical random bit flips on output qubits
✓ Correct Answer:
The correct answer is 4) Probabilistically applying Pauli errors and a distinguishability operator, tracking stabilizers and minimal internal state per photon.
📚 Reference Text:
the cases in which the photon in mode 2i(respectively 2i+ 1) is distinguishable. We will also consider a second-order pair error , in which the input photons in both modes 2iand2i+ 1are distinguishable with the same internal state η1=|ψ1⟩⟨ψ1|withTr(η0η1) = 0 . We note that in Ref. 20, the output state of the Bell state generator was very nicely calculated by entirely tracing out the internal states of all photons. Here we choose not to trace out the internal degrees of freedom entirely, instead giv- ing a framework in which distinguishability errors on the input photons may be mapped to a combination of Pauli and distinguishability errors on the output state. We briefly introduce some notation for this purpose. For an operator A, letPA(γ) =1 2(γ+AγA†). Also let Dibe an operator that acts on modes 2i,2i+ 1 bya† 2i(η0)|⃗0⟩ 7→ a† 2i(η1)|⃗0⟩, a† 2i+1(η0)|⃗0⟩ 7→ a† 2i+1(η1)|⃗0⟩. This operator is simply for convenience of notation, and allows us to write partially- distinguishable mixed states by starting with the ideal state |B+⟩⟨B+|and applying appropriate operators. Theorem 4. Suppose the n-GHZ state generation protocol heralds success with measurement pattern ⃗ m= (m0, . . . , m 2n−1). Assume that all input photons except for those in input modes 2i,2i+ 1are ideal, with internal state η0=|ψ0⟩⟨ψ0|. 1. Suppose that input photons 2i,2i+ 1are in state (16) above. After dual-rail projection, the output state is 1 2PXi(PZi(|B+⟩⟨B+|) +Di(|B+⟩⟨B+|)). (17) 2. Suppose we have a pair error on pair i, so that input photons 2i,2i+ 1 are in the same internal state η1, fully distinguishable from the ideal internal state η0. Then the output state is PZi(Di(|B+⟩⟨B+|)), with no dual-rail projection required. Remark 5. We note that these results suggest a paradigm for computing the effect of low-order distinguishability errors on much larger circuits. Suppose we have a large circuit whose subroutines are n-GHZ state generation, single-qubit Clifford operations, Type I fusions, and k-GHZ state analyzers (for varying n, k). Also suppose that the distinguishability error rates are sufficiently low that low-order error approximations are reasonable. Theorem 4 shows that, under these circumstances, erroneous n-GHZ state generation may be simulated by taking the ideal state and probabilistically applying Pauli errors and an operator Dithat converts ideal photons to fully distinguishable ones. This sampled state may then be fed into the later unitary operations, fusion gates, etc.; it is straightforward to see how Pauli errors propagate through the circuit, as with standard Pauli frame calculations, and the results of Sec. II D 1 show how distinguishability errors interact with each operation. This implies the possibility of a generalized stabilizer-type simulation, in which one tracks both the stabilizers of the simulated state and, for each photon, a minimal amount of information about its internal state. This, of course, will be simplest if one uses an appropriately simple distinguishability error model, such as those discussed in Sec. II D 3 below. Further, the assumption that the errors are low-order is essential: otherwise, multiple interacting errors could “cancel,
Question 87multiple-choice
Quantum neural networks leverage mathematical structures to efficiently represent and manipulate complex quantum data. Clifford algebras are particularly valued in this domain for their ability to encode geometric relationships and facilitate quantum computations.
Which of the following statements accurately describes a key advantage of using Clifford algebras, represented via Pauli matrices, in the design of quantum neural networks?
1) They guarantee the universal approximation of all classical functions without the need for entanglement.
2) They exclusively facilitate linear transformations, limiting their applicability to simple datasets.
3) They enable the implementation of irreversible activation functions required for quantum learning.
4) They provide a natural framework for representing multidimensional data and capturing geometric properties crucial for quantum computing.
5) They replace the need for unitary operations in quantum algorithms by offering non-unitary alternatives.
6) They ensure that quantum neural networks operate entirely without data entanglement.
7) They directly generalize Boolean logic gates for classical neural network architectures.
✓ Correct Answer:
The correct answer is 4) They provide a natural framework for representing multidimensional data and capturing geometric properties crucial for quantum computing..
📚 Reference Text:
Title: Clifford Algebras, Quantum Neural Networks and Generalized Quantum Fourier Transform Year: 2022 Paper ID: 7ab54e6a2b2a0908f4e2ea789bf3a9cb3972eda7 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/7ab54e6a2b2a0908f4e2ea789bf3a9cb3972eda7 Abstract: We propose models of quantum perceptrons and quantum neural networks based on Clifford algebras. These models are capable to capture geometric features of classical and quantum data as well as producing data entanglement. Due to their representations in terms of Pauli matrices, the Clifford algebras seem to be a natural framework for multidimensional data analysis in a quantum setting. In this context, the implementation of activation functions, and unitary learning rules are discussed. In this scheme, we also provide an algebraic generalization of the quantum Fourier transform containing additional parameters that allow performing quantum machine learning based on variational algorithms. Furthermore, some interesting properties of the generalized quantum Fourier transform have been proved.
Question 88multiple-choice
Quantum algorithms frequently employ advanced measurement strategies and circuit constructions to solve computationally hard problems. The relationship between efficient quantum circuit implementation and classical computational complexity is a central consideration in quantum algorithm design.
Which statement accurately describes the implication of efficient quantum circuit implementation for a rank-one POVM associated with the subset sum problem?
1) Efficient implementation guarantees the subset sum problem becomes polynomial-time classically solvable.
2) Efficient implementation allows arbitrary sampling from all NP-complete problems.
3) Efficient implementation implies the rank of the Hilbert space must be exponentially large.
4) Efficient implementation requires all blocks of the unitary transformation to be uniquely determined.
5) Efficient implementation is always possible for any instance regardless of structure.
6) Efficient implementation of the optimal POVM enables efficient quantum sampling from subset sum solutions, effectively solving an NP-complete problem.
7) Efficient implementation makes Neumark’s theorem unnecessary for realizing POVMs.
✓ Correct Answer:
The correct answer is 6) Efficient implementation of the optimal POVM enables efficient quantum sampling from subset sum solutions, effectively solving an NP-complete problem..
📚 Reference Text:
perform this measurement in a particular way, first measuring the labelxand then performing the povm {Ex j}j∈ZNconditioned on that label. Note that each Ex jis rank one, so that the povm {Ex j}j∈ZNfor fixed x∈Zk Nis refined into one-dimensional subspaces, removing much of the freedom in the implementation of the original povm {Ej}. For any given x∈Zk N, we consider the implementation of the povm {Ex j}j∈ZNby anx-dependent quantum circuit followed by a measurement in the computational basis to give the outcome j. In general, this circuit and measurement will act on a larger Hilbert space than is required to hold the original input. The quantum circuit will then correspond to some unitary operation Uxon the larger space. Without loss of generality, we can assume that the final measurement is in a basis {|j/angbracketright}such that the values j∈ {0,1,...,N −1}indicate the measurement outcome Ej. According to Neumark’s theorem [43, Chap. 9.6], the unitary operator Uxhas the block form Ux=/parenleftbigg VxAx BxCx/parenrightbigg (65) where Vx:=1√ N/summationdisplay j,q∈ZNω−jq|j/angbracketright/angbracketleftSx q| (66) is a fixed ( N×2k)-dimensional matrix whose columns are the (subnor- malized) vectors corresponding to the rank one povm elements {Ex j}, and Ax,Bx,Cxare arbitrary up to the requirement that Uxis unitary. It is convenient to perform a Fourier transform on the left, i.e., on the index j 18 DAVE BACON, ANDREW M. CHILDS, AND WIM VAN DAM (overZN, for the relevant values j∈ {0,1,...,N −1}), giving a unitary operator ˜Ux=/parenleftbigg˜VxAx ˜BxCx/parenrightbigg (67) with ˜Vx:=1 N/summationdisplay j,p,q∈ZNωj(p−q)|p/angbracketright/angbracketleftSx q| (68) =/summationdisplay p∈ZN|p/angbracketright/angbracketleftSx p|. (69) Clearly,Uxcan be implemented efficiently if and only if ˜Uxcan be imple- mented efficiently. Therefore, if we have an efficient quantum circuit for the transformation (70) |p,x/angbracketright /mapsto→/braceleftBigg |Sx p,x/angbracketrightηx p>0 |ψx p/angbracketrightηx p= 0 where |ψx p/angbracketrightis any state allowed by the unitarity of ˜Ux(i.e., if we can effi- ciently quantum sample from subset sum solutions for legal inputs), then by running this circuit in reverse, we can efficiently implement ˜Ux, and hence the measurement. Conversely, given the ability to implement the optimal povm by the mea- surement of xfollowed by an efficient implementation of Ux, we can solve the subset sum problem. By running the quantum circuit for ˜Uxin the reverse direction, we can efficiently implement the transformation (70). Suppose we are trying to solve the subset sum problem for a legal instance ( x,t). Using (70), we can produce the state |Sx t/angbracketright, which upon measurement gives a uni- formly random subset of xsumming to t. On the other hand, if the instance is not legal, then we can easily check that the output does not correspond to a subset of xsumming to t. If we could efficiently implement the unitary operation ˜Uxfor anyk= poly(logN), then we could solve the subset sum problem efficiently even in the worst case. Since the subset sum decision problem is NP-complete, such an implementation seems unlikely. However, for the purpose of solving thedhsp , it is sufficient for each Nto consider a specific k=νlogN(with ν >1, according to Theorem 2) and
Question 89multiple-choice
Quantum reductions are a foundational tool for understanding the relative hardness of cryptographic problems, especially in the development of secure post-quantum cryptosystems. Techniques such as the quantum Fourier transform and custom quantum circuits are often employed to construct reductions between problems like EDCP and LWE.
Which operation is essential in both quantum reductions between the Extended Dihedral Coset Problem (EDCP) and Learning with Errors (LWE) for extracting structural information from quantum superpositions?
1) Quantum Fourier Transform (QFT)
2) Grover's search algorithm
3) Quantum phase estimation
4) Quantum error correction
5) Classical post-processing only
6) Quantum teleportation
7) Quantum amplitude amplification
✓ Correct Answer:
The correct answer is 1) Quantum Fourier Transform (QFT).
📚 Reference Text:
building quantum reductions between them. The EDCP problem over D Nis specified as follows: Given /lscriptmany registers in a normalized state corresponding to /summationdisplay j∈Ze−π|j|2 r2|j,(xi+j·s)mod N/angbracketright where xi∈Zn N(i=1,...,/lscript ), and s∈Zn Nis fixed, the objective of EDCP is to find the secret value s. (1) Quantum reduction from LWE to EDCP. An instance of LWE problem over the lattice L(A),A∈Zm×n q , can be reduced to an instance of EDCP problem over the dihedral group DN,N=2n, according to the following quantum steps (Fig. 6): •Step 1 Initialize the four registers with required qubits |ϕ1/angbracketright=| 0/angbracketright|0/angbracketright|0/angbracketright|0/angbracketright •Step 2 Perform QFT on the second register (normalization omitted) |ϕ2/angbracketright=/summationdisplay s∈Znq|0/angbracketright|s/angbracketright|0/angbracketright|0/angbracketright 123 Quantum algorithms for typical hard problems… Page 21 of 26 178 •Step 3 Apply GR02 algorithm [ 81] in the first register, which is a quantum process to create a superposition state according to given probability distribution |ϕ3/angbracketright=/summationdisplay s∈Znq(/summationdisplay j∈Zρr(j)|j/angbracketright)|s/angbracketright|0/angbracketright|0/angbracketright •Step 4 Suppose that the quantum circuit Uf Uf|j/angbracketright|s/angbracketright|0/angbracketright→| j/angbracketright|s/angbracketright|As−jbmod q/angbracketright is at hand. Apply U fon the first three registers |ϕ4/angbracketright=/summationdisplay s∈Znq,j∈Zρr(j)|j/angbracketright|s/angbracketright|As−j·As0−je0/angbracketright|0/angbracketright =/summationdisplay s∈Znq,j∈Zρr(j)|j/angbracketright|s+js0/angbracketright|As−je0/angbracketright|0/angbracketright •Step 5 Further, suppose that the quantum circuit Ug Ug|x/angbracketright|0/angbracketright→| x/angbracketright|x/z−wmod¯q/angbracketright is at hand, where ¯q=q/z=c. Apply U gon the last two registers, and we get |ϕ5/angbracketright=/summationdisplay s∈Znq,j∈Zρr(j)|j/angbracketright|s+j·s0/angbracketright|As−je0/angbracketright|g(As−je0)/angbracketright •Step 6 Measure the fourth register and discard it |ϕ6/angbracketright=/summationdisplay j∈Zρr(j)|j/angbracketright|s+j·s0/angbracketright|As−je0/angbracketright •Step 7 Apply Ufto the first three registers, the third register gives 0 and discard it, and the state is of the form |ϕ7/angbracketright=/summationdisplay j∈Zρr(j)|j/angbracketright|s+j·s0/angbracketright •Step 8 Repeat the above procedure /lscripttimes, and we obtain /lscriptmany EDCP states with probability (1−1 k)m/lscript |ϕEDCP/angbracketright={/summationdisplay j∈Zρr(j)|j/angbracketright|s+j·s0/angbracketright}k≤/lscript where xk∈Znq. 123 178 Page 22 of 26 J. Suo et al. Fig. 7 From EDCP to LWE [80].FZnqis Fourier transform over Znq,a n d UED CP is the output stateρt(j)|j FZnq ULWE |x+j·s0FZnq ˆa (2) Quantum reduction from EDCP to LWE. The reverse quantum reduction from EDCP to LWE is given below (Fig. 7). •Step 1 Prepare the input state |ϕ1/angbracketright=/summationdisplay j∈Zρr(j)|j/angbracketright|x+j·s0mod q/angbracketright •Step 2 Apply QFT on the second register |ϕ2/angbracketright=/summationdisplay a∈Znq/summationdisplay j∈Zω/angbracketleft(x+j·s0),a/angbracketright q ·ρr(j)|j/angbracketright|a/angbracketright where ωq=e2πi q •Step 3 Measure the second register and obtained ak, omitting global phase ω/angbracketleftx,/hatwidea/angbracketright q |ϕ3/angbracketright=/summationdisplay j∈Zωj·/angbracketleft/hatwidea,s0/angbracketright q ·ρr(j)|j,/hatwidea/angbracketright •Step 4 Apply QFT on the first register |ϕ4/angbracketright=/summationdisplay b∈Zq/summationdisplay j∈Zqωj·(/angbracketleft/hatwidea,s0/angbracketright+b) q ·ρr(j)|b/angbracketright Using Poisson summation formula to reorganize |ϕ4/angbracketright, then |ϕ4/angbracketright=/summationdisplay e∈Zρ1 2/parenleftbigge q/parenrightbigg |−/hatwidea,s0/angbracketright+emod q/angbracketright •Step 5 Measure the first register, and we can obtain an LWE sample |ϕLWE/angbracketright=(−/hatwidea,/angbracketleft−/hatwidea,s0/angbracketright+ek) 9 Conclusion With the rapid development of quantum computing, it broke through the defense line of the classic cryptosystems, which makes the post-quantum cryptography become the frontier of research. In order to search the novel cryptography which is resistant to 123 Quantum algorithms for typical hard problems… Page 23 of 26 178 quantum attack, it is of great necessity to conduct a systematical analysis of the quan- tum algorithms that could solve the typical hard problems. In this paper, we start from the typical hard problems: integer factorization problem, discrete logarithmic problemand dihedral hidden subgroup problems in the public-key cryptosystem (respectively, RSA, ElGamal,
Question 90multiple-choice
Hopf algebras constructed from quivers and categories are fundamental structures in modern algebra, with rich interactions between categorical properties, module categories, and specialized algebraic relations. The interplay between morphism composition, tensor products, and specialization at roots of unity yields diverse algebraic frameworks with applications in representation theory and quantum groups.
Which property ensures that non-composable morphisms in the associative algebra Kc constructed from a category are represented by orthogonal elements, and what is the algebraic consequence of this?
1) Non-trivial kernel in the comultiplication, causing all morphisms to act identically
2) Existence of a two-sided ideal generated by identity morphisms, making all compositions non-zero
3) Direct product decomposition of Kc into simple modules, ensuring full reducibility
4) Orthogonality of non-composable morphisms, meaning their product is zero and respects the underlying categorical structure
5) Coassociativity of the tensor product, making all morphism pairs composable
6) Specialization at roots of unity, causing certain paths to collapse to the unit
7) Action of the free group by translations, associating every morphism to a unique matrix entry
✓ Correct Answer:
The correct answer is 4) Orthogonality of non-composable morphisms, meaning their product is zero and respects the underlying categorical structure.
In group theory, central extensions and Burnside groups are central topics in the study of infinite groups and group varieties. Understanding the relationships between centers, quotients, and cohomological invariants reveals deep structural properties of groups.
Which construction allows any countable abelian group to be realized as the center of a group whose quotient is isomorphic to a free Burnside group of given rank and odd period?
1) A central extension using a countable abelian group as the center and a free Burnside group of odd period as the quotient
2) A direct product of a free Burnside group and a cyclic group of finite order
3) An amalgamated free product of abelian groups and finite cyclic groups
4) A semidirect product of a Burnside group with an infinite symmetric group
5) A wreath product of a countable abelian group with a finite cyclic group
6) A quotient of an infinite cyclic group by a finite abelian subgroup
7) An HNN extension with a free Burnside group as the base group
✓ Correct Answer:
The correct answer is 1) A central extension using a countable abelian group as the center and a free Burnside group of odd period as the quotient.
📚 Reference Text:
Title: Central extensions of free periodic groups Year: 2018 Paper ID: e5e8ce8bc6b5a8a1a22e39df90605bd551d9b7d8 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/e5e8ce8bc6b5a8a1a22e39df90605bd551d9b7d8 Abstract: It is proved that any countable abelian group can be embedded as a centre into a -generated group such that the quotient group is isomorphic to the free Burnside group of rank 1$?> and of odd period . The proof is based on a certain modification of the method that was used by Adian in his monograph in 1975 for a positive solution of Kontorovich’s famous problem from the Kourovka Notebook on the existence of a finitely generated noncommutative analogue of the additive group of rational numbers with any number 1$?> of generators (in contrast to the abelian case). More precisely, he proved that the desired analogues in which the intersection of any two non-trivial subgroups is infinite, can be constructed as a central extension of the free Burnside group , where 1$?>, and is an odd number, using the infinite cyclic group as its centre. The paper also discusses other applications of the proposed generalization of Adian’s technique. In particular, the free groups of the variety defined by the identity and the Schur multipliers of the free Burnside groups for any odd are described. Bibliography: 14 titles.
Question 92multiple-choice
In representation theory and symplectic geometry, moment polytopes encode geometric and algebraic information about group actions on vector spaces. The structure of these polytopes is closely related to the facets defined by certain elements associated with the representation.
Which condition must be satisfied by a Ressayre element (H, z) to ensure that the associated matrix constructed from root operators and weight vectors is meaningful for describing a non-trivial facet of the moment polytope?
1) The matrix must be symmetric and positive definite.
2) The determinant of the matrix must be zero.
3) The matrix must have strictly positive eigenvalues.
4) The trace of the matrix must be equal to the dimension of the representation.
5) The determinant of the matrix must be non-zero.
6) The matrix must be diagonalizable over the complex numbers.
7) The rank of the matrix must be less than that of the weight space.
✓ Correct Answer:
The correct answer is 5) The determinant of the matrix must be non-zero..
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For any H2Gandz2Z, we now define the following three sub(multi)sets: (H=z) =f'2(M) :'(H) =zg; (H 0for any of the simple coroots h . Equation (11) implies that any non-trivial facet is necessarily given by a Ressayre element. 3.4.MomentPolytope is in coNP. A problem instance for MomentPoly- topeis given by a quadruple (G;M;;k )encoded as described in section 3.1 above. We now describe a polynomial-time algorithm that takes as input the problem instance MEMBERSHIP IN MOMENT POLYTOPES IS IN NP AND CONP 15 together with a certificate that consists of a triple (H;z;p ), whereH2G,z2Z, andp2Z#(H=z). Here,His specified as an integer vector with respect to the bases fixed at the beginning of section 3. The algorithm proceeds as follows: We first check the conditions in Definition 11 to verify that (H;z)is a Ressayre element for G(M): 1.Admissibility: There are #(M) =dimM=O(hMi)weights, each living in a space of dimension dimT=O(hGi). According to Lemma 8, we can compute the multiset of weights (M)in polynomial time. For each weight '2(M), we can determine if '2(H=z)by verifying that '(H) =z, which amounts to evaluating an inner product in ZdimT. Thus we can in polynomial time determine (H=z)and compute the rank of the polynomial-size matrix with columns ' 1 for'2(H=z). The element (H;z)is admissible if and only if the rank is equal to dimT. 2.Trace condition: As there are no more than O(hGi2)negative roots and dimM=O(hMi)weights, each of which lives in a space of
Question 93multiple-choice
In mathematical physics and representation theory, understanding the structure of Lie group representations and their reducibility is essential for analyzing symmetries in quantum systems. Compact Lie groups, operators, and commutation relations play foundational roles in this analysis.
Which property of compact Lie groups is essential for the validity of Haar averaging, used to ensure that every finite-dimensional representation is equivalent to a unitary representation?
1) The group has a non-Abelian structure
2) The group has an infinite-dimensional center
3) The group contains raising and lowering operators
4) The group is simple but not semisimple
5) The group admits only reducible representations
6) The group has a discrete topology
7) The group has finite volume under its natural topology (compactness)
✓ Correct Answer:
The correct answer is 7) The group has finite volume under its natural topology (compactness).
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be true forSU(2)): there are counterexamples for all SU(d) withd>3, whereinthesamedimensionalvectorspacemay have several inequivalent representations. Secondly, we see the powerful strategy for detecting equivalent representa- tions (partially aforementioned after Example 9), at play: diagonalize the representatives of Zin both representations (here, the maps Z 1+1 Zon Sym2(C2)andadZ()on su(2)),then use other operators and commutation relations to move between these eigenbases . Let us see how this strategy played out. We began the tensor example by diagonalizing the Zrepresentative in the tensor rep by choosing the basis j11i;j10i+j01i;j00i. In this basis, the Zrepresentative has respective eigenval- ues 2, 0, and -2. Then, the raising and lowering operators map between these eigenspaces, which looks like “rais- ing” or “lowering” the eigenvalue. It becomes clear that we can map any of these eigenspaces to each other by raising and/or lowering operators: thus, we have an irreducible subspace. When we got to the adjoint representation and saw thatadZ()had the same eigenvalues, this suggested a similar strategy should work. There is nothing inherently special about the Zoperator here–it just so happens to be fairly easy to diagonalize in both cases. This generalizes to more complicated Lie algebras by simultaneously diagonal- izing a commuting collection of operators (called a Cartan subalgebra ) and cleverly tracking their eigen-information using so-called “weights”. In practice, diagonalization in a given representation may be highly nontrivial. Checking the reducibility of any acquired invariant subspace may also be a difficult task, and often requires either problem specific insight or any variety of representation theoretic tools. It cannot however be understated how crucial this strat- egy is. It undergirds the classification of semisimple Lie al- gebras. Italsoelegantlyframestheanalysisofthequantum harmonic oscillator: there, we diagonalize the self-adjoint number operator ayato obtain eigenstates, and then use 25 the canonical commutation relations [x;p] =i~combined withpositivityof ayatomovebetweeneigenstatesvia“rais- ing and lowering” ladder operators. This is well trodden and exposited ground, and every reference we list contains thiscomputation. InHall[51], thiscomputationappearsin the section “Representations of sl2(C)” (recall that as com- plexified Lie algebras, su(2)=sl2(C), and so they have the same representation theory). Theorem 5 (Weyl’s unitary trick for complete reducibil- ity).LetGbe a compact Lie group. Then every finite di- mensional representation of Gis equivalent to a unitary representation, and by Theorem 4, completely reducible. While we omit the (brief) proof here, there are two im- portant messages from it to be mentioned: •The equivalence is established by performing a Haar averaging procedure15to the dot product h;ion V, much akin to that of the twirl operator (we will visit these in Sections VIIA, VIIB). These sorts of Haar averaging methods are of great importance across quantum computation and representation the- ory alike, since the Haar measure is the most “natu- ral” probability distribution on Lie groups. •Compactness is critical here: without it, the Haar measure of our Lie group jGjmay not be finite and the averaging procedure is nonsensical. The final result we mention is perhaps the most critical weapon any researcher needs in their arsenal before using representation theory to fully tackle QML
Question 94multiple-choice
In quantum information theory, symmetries are deeply connected to the structure and implementation of quantum channels. Programmable quantum processors are devices used to flexibly realize families of quantum operations, often leveraging properties such as group representations and associated mathematical tools.
Under what condition does a programmable quantum processor implementing covariant quantum channels admit a measure-and-prepare form, simplifying its structure?
1) When the symmetry group is non-abelian and acts reducibly on the input space
2) When the tensor representation has only one irreducible component
3) When the commutant of the tensor representation is abelian
4) When the dimension of the program space equals the dimension of the input Hilbert space
5) When the symmetry group contains only unitary elements
6) When the Choi-Jamiołkowski state is maximally mixed
7) When every representation of the group is reducible
✓ Correct Answer:
The correct answer is 3) When the commutant of the tensor representation is abelian.
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also hold for finite groups. Symmetries are of fundamental importance in physics, since they give rise to conserved quantities via Noether’s theorem [33]. In open systems, these symmetries arise as covariant quantum channels and are studied using tools from quantum information theory [11, 29]. From a practical point of view, symmetries often simplify problems and break the curse of dimen- sionality, thus making them amenable to rigorous analysis. For these reasons, covariant quantum channels appear in many different settings, such as channel discrimination, ca- pacities and communication tasks (see Ref. [30] and the references therein). Since this is a special set, the question arises whether those channels can be implemented exactly by a programmable quantum processor. In this article, we show that an exact implementation is possible if the group acts irreducibly on the channel input. While we prove upper bounds on the program dimension for general representations U, we focus on the case in which U is irreducible. As any representation can be decomposed into irreducible ones, it is natural to start studying this scenario before investigating more general representations in future work. In Section 2, we present some preliminaries and our notation. We consider exact pro- grammability of group-covariant channels in Section 3, where we first look at a method based on extreme points in Subsection 3.1. We show that processors have a particularly simple measure-and-prepare form if and only if the commutant of the tensor representation is abelian. This yields a program dimension equal to the number of irreducible representa- tions occurring in the direct sum decomposition of the tensor representation in Corollary 19. Subsection 3.2 discusses the structure of the commutant of the tensor representa- tion in more detail. We give a different construction of covariant programmable quantum processors based on teleportation in Subsection 3.3. This construction is subsequently concatenated with a compression map which allows us to utilize the special structure of the Choi-Jamiołkowski states corresponding to the covariant channels. This leads to The- orem 25, where we show that we obtain a program dimension of at most the sum of the dimensions of the blocks occurring in the structure of the Choi-Jamiołkowski states. After the analysis of exact programmability, we consider an approximate version thereof (see Section 4). First, we provide approximate upper bounds in the case of arbitrary repre- sentationsU,Vin Proposition 29. They are in general worse than the exact bounds in Theorem 25, but they apply more generally. This result is the only one in which we con- sider arbitrary representations Uinstead of irreducible ones. In Theorem 31, we provide lower bounds on the program dimension of approximate covariant quantum processors. In particular, this shows that the construction in Theorem 25 is optimal for the exact case. 2 Preliminaries We use the following notation: Let d1,d2∈N. We denote the set of bounded linear operatorsH1→H 2withd1- andd2-dimensional Hilbert spaces H1andH2byB(H1)and B(H2), respectively. The set of all d1-dimensional density operators is D(H1) ={ρ∈B(H1)|ρ≥0,tr(ρ) = 1}, Accepted in Quantum2021-06-18, click title to verify. Published under
Question 95multiple-choice
Quantum Markov semigroups (QMS) play a fundamental role in modeling the dynamics of open quantum systems, particularly when symmetries are present via group representations. The relationship between classical and quantum processes is often explored through the structure of fixed points and their connection to operator algebras.
In the construction of a quantum Markov semigroup on operators over a Hilbert space H, arising from a projective unitary representation u(g) of a group G, which set precisely characterizes the fixed points of the semigroup?
1) All diagonal operators in B
2) The center of B
3) Operators invariant under all *-automorphisms of B
4) The set of all positive operators in B
5) The commutant of the projective representation, i.e., all X in B satisfying X u(g) = u(g) X for all g in G
6) The set of trace-class operators in B
7) Operators with support orthogonal to u(g) for all g in G
✓ Correct Answer:
The correct answer is 5) The commutant of the projective representation, i.e., all X in B satisfying X u(g) = u(g) X for all g in G.
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of completness. We have, for any x∈B(H), ○Tt(x)(g) =u(g)∗/integral.dispGkt(h−1)u(h−1)xu(h)dG(h)u(g) =/integral.dispGkt(gg−1h−1)u((hg)−1)xu(hg)dG(h) =/integral.dispGkt(gh−1)u(h−1)xu(h)dG(h) =(St⊗idB(H))((x))(g): From the invariance of G, one can also easily verify that any QMS (Tt)t≥0transferred from (St)t≥0is doubly stochastic :T† t(d−1 HIH)=d−1 HIHfor anyt≥0. On the other hand, the reversibility of (St)t≥0is transferred to the QMS (Tt)t≥0: Lemma II.2. Assume that the Markov semigroup (St)t≥0 is reversible, or equivalently that kt(g)=kt(g−1)for any g∈G. Then any QMS (Tt)t≥0transferred from (St)t≥0 is self-adjoint with respect to d−1 HIH. Proof. The result follows from the simple calculation: /uni27E8x;Tt(y)/uni27E9HS =/integral.dispGkt(g−1)Tr(x∗u(g)∗yu(g))dG(g) =/integral.dispGkt(g−1)Tr((u(g)xu(g)∗)∗y)dG(g) =/integral.dispGkt(g−1)Tr((u(g)∗xu(g))∗y)dG(g) =/uni27E8Tt(x);y/uni27E9HS; where the third line follows from the identity kt(g)= kt(g−1)for allg∈G. Sinced−1 HIHis an invariant state of (Tt)t≥0, the setNfixof fixed points is an algebra (see [9], [12], Theorem 6.12 of [10]), and is characterized as the commutant of the projective representation (see Theorem 6.13 of [10]): Nfix={∈B(H)/divides.alt0∀g∈G∶u(g)=u(g)}≡u(G)′: (5) By definition, it is also the algebra of fixed points of the∗-automorphisms x/uni21A6u(g)∗xu(g),g∈G. This implies that the following commuting diagram holds: 3 B(H) Nfix L∞(G;B(H)) B(H):Efix EG HereB(H)in the lower right corner has to be understood as the subalgebra of constant value functions on Gwith value in B(H). In practice, we will only consider situations where the classical Markov semigroup (St)t≥0isprimitive , that is,Gis the unique invariant distribution and furthermore Stf/leftrightline→ t→+∞EG[f]=/integral.dispGf(g)dG(g): This does not imply that (Tt)t≥0is also primitive, however, it will always be ergodic as defined in Equation (1). We now turn our attention to two special cases of the above construction. In both cases, we explicitly construct the Lindblad generator of the QMS. B. Diffusion Given a Riemannian manifold M, aHörmander system onM is a set of vector fields V= {V1;:::;Vm}such that, at each point p∈M, there exists an integer Ksuch that the iterated commutators /bracketleft.alt1Vi1;[Vi2;/uni22EF[Vik;⋅]]/bracketright.alt,k=1;:::;K , generate the tangent space TpM. Specializing to the case of a Lie groupG, a Hörmander system V={V1;:::;Vm}can more simply be defined as a set of vectors in the Lie algebra, i.e. the tangent space at the neutral element e, such that for some K∈Nthe iterated commutators of order at most Kspan the whole tangent space. For fixed j∈{1;:::;K}, we find a geodesic gj(t)withgj(0)=e such that for any f∈C1(G) Vj(f)(h)=d dtf(gj(t)h)/divides.alt3 t=0: This leads to the corresponding left invariant classical generator LV∶=−/summation.disp jV2 j: (6) The generator LVgenerates a Markov semigroup Pt=e−tLVonL∞(G). Whenever Vis a basis for the Lie algebra, then LV≡−is the negative of the Laplacian and (Pt)t≥0is called the heat semigroup . Since the semigroup commutes with the right action of the group it is implemented by a right-invariant convolution kernel as in Equation (2) and it is reversible with respect to the Haar measure. Next, considering a projective representation g/uni21A6 u(g)ofGon some finite dimensional Hilbert space H, we want to find the Lindblad generator of the QMS defined by Equation (3). We first observe that, for fixed j∈{1;:::;K}and given the geodesic gjassociated tothe vector field Vj,u(gj(t))is a one-parameter family of unitaries and hence d dtu(gj(t))/divides.alt3 t=0=iaj (7) whereaj∈B(H)is self-adjoint. This implies that, for anyx∈B(H), (Vj⊗idB(H)○(x)(g) =d dt(x)(gj(t)g)/divides.alt3
Question 96multiple-choice
Transformation monoids are algebraic structures that generalize permutation groups by considering actions of monoids on sets, with important applications in combinatorics, representation theory, and theoretical computer science. Primitivity and orbital digraphs are key concepts for understanding the symmetry and structure of these actions.
In the context of finite transformation monoids, which statement correctly describes the connection between primitivity and orbital digraphs as established by the monoid version of Higman's theorem?
1) A transformation monoid is primitive if and only if all its non-trivial orbital digraphs are connected.
2) A transformation monoid is primitive if and only if all its orbital digraphs are acyclic.
3) A transformation monoid is primitive if and only if every element acts as a permutation.
4) A transformation monoid is primitive if and only if its orbital digraphs have minimal rank.
5) A transformation monoid is primitive if and only if its transformations preserve every partition of the set.
6) A transformation monoid is primitive if and only if its orbital digraphs are bipartite.
7) A transformation monoid is primitive if and only if there exists a transformation mapping every element to a fixed point.
✓ Correct Answer:
The correct answer is 1) A transformation monoid is primitive if and only if all its non-trivial orbital digraphs are connected..
📚 Reference Text:
Title: A Theory of Transformation Monoids: Combinatorics and Representation Theory Year: 2010 Paper ID: 2c1ab91e10e336edeef6301d6dac1d0afabaafce Source: semantic-scholar URL: https://www.semanticscholar.org/paper/2c1ab91e10e336edeef6301d6dac1d0afabaafce Abstract: The aim of this paper is to develop a theory of finite transformation monoids and in particular to study primitive transformation monoids. We introduce the notion of orbitals and orbital digraphs for transformation monoids and prove a monoid version of D. Higman's celebrated theorem characterizing primitivity in terms of connectedness of orbital digraphs. A thorough study of the module (or representation) associated to a transformation monoid is initiated. In particular, we compute the projective cover of the transformation module over a field of characteristic zero in the case of a transitive transformation or partial transformation monoid. Applications of probability theory and Markov chains to transformation monoids are also considered and an ergodic theorem is proved in this context. In particular, we obtain a generalization of a lemma of P. Neumann, from the theory of synchronizing groups, concerning the partition associated to a transformation of minimal rank.
Question 97multiple-choice
The hidden subgroup problem (HSP) is fundamental in quantum computing, underpinning many algorithms that achieve exponential speedups. Solving HSP in nonabelian groups, such as semidirect products of cyclic groups, presents unique challenges and opportunities for algorithmic advancement.
Which of the following statements correctly identifies a key technical strategy used by a polynomial-time quantum algorithm for the HSP over the group $\mathbb{Z}_{p^r} \rtimes \mathbb{Z}_{q^s}$, where $p$ and $q$ are odd primes?
1) Employing the nonabelian quantum Fourier transform to directly identify all subgroup structures
2) Utilizing quantum amplitude amplification to distinguish between normal and non-normal subgroups
3) Applying Grover’s search to enumerate hidden subgroups exhaustively
4) Reducing the problem to solving instances of the graph isomorphism problem
5) Using the abelian quantum Fourier transform and reducing the problem to finding cyclic subgroups
6) Implementing quantum walks to traverse the group’s Cayley graph
7) Leveraging classical algorithms for abelian subgroup decomposition
✓ Correct Answer:
The correct answer is 5) Using the abelian quantum Fourier transform and reducing the problem to finding cyclic subgroups.
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Title: Solution to the Hidden Subgroup Problem for a Class of Noncommutative Groups Year: 2011 Paper ID: 757dafd558edcf123a164111a26ad780c56d99e1 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/757dafd558edcf123a164111a26ad780c56d99e1 Abstract: The hidden subgroup problem (HSP) plays an important role in quantum computation, because many quantum algorithms that are exponentially faster than classical algorithms can be casted in the HSP structure. In this paper, we present a new polynomial-time quantum algorithm that solves the HSP over the group $\Z_{p^r} \rtimes \Z_{q^s}$, when $p^r/q= \up{poly}(\log p^r)$, where $p$, $q$ are any odd prime numbers and $r, s$ are any positive integers. To find the hidden subgroup, our algorithm uses the abelian quantum Fourier transform and a reduction procedure that simplifies the problem to find cyclic subgroups.
Question 98multiple-choice
Quantum dot-cavity systems are central to advancing photonic quantum computing, enabling high-fidelity quantum logic gates and scalable implementations of algorithms such as the discrete quantum Fourier transform. Key physical parameters determine the efficiency and reliability of photon-based quantum operations.
Which parameter regime is essential for ensuring strong coupling and high-fidelity interaction between photons and quantum dot-cavity systems in the implementation of deterministic quantum logic gates?
1) g ≫ (κ, γ) and κs ≪ κ
2) g ≪ (κ, γ) and κs ≫ κ
3) κ ≫ g and γ ≫ κs
4) γ ≫ g and κ ≪ κs
5) κs ≫ κ and g ≈ γ
6) g ≈ κ and γ ≈ κs
7) κ ≈ γ and g ≪ κs
✓ Correct Answer:
The correct answer is 1) g ≫ (κ, γ) and κs ≪ κ.
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trapped atoms used as qubits. Compared with the previous schemes15,17,21,22,25,27,31–33, our DQFT scheme consists of the QD-cavity systems (nonlinearly optical devices) to implement the CRk operations as the basic modules for the deterministic perfor - mance and simple expansion into multi-qubit DQFT. Moreover, in our scheme, the flying photons play the roles of qubits transferring quantum information, and QDs are the ancillary systems to efficiently perform the CRk operations for the applications of various quantum computations and algorithms. Also, In view of the experimental realization, for the high fidelity (condition of g ≫ (κ, γ) and κ s ≪ κ) of the interaction between a photon and the QD-cavity system, researchers have proposed a variety of experimental techniques, including the following: Rosenblum et al . 68 obtained the coupling strength as g /(κ + κs) ≈ 0.5 in a micropillar cavity (d = 1.5 μm) with a quality factor Q = 8800, and for Q = 40000, the coupling strength could be increased to g /(κ + κs) ≈ 2.474. Furthermore, Arnold et al .76 enhanced the quality factor, Q = 215000 (κ ≈ 6.2 μeV), in order to acquire a small side-leakage rate. Reitzensteina et al .77 demonstrated two methods (the etching process and improving the sample growth) to reduce the side-leakage rate, κs/κ, in an In0.6Ga0.4As optical cavity having g/(κ + κs) ≈ 2.4 and Q = 40000. Thus, the QD-cavity systems in our CRk gate can be reliably operated with high fidelity, according to the results of our analysis presented in Sec. 4. Moreover, in order to ensure the reliable interaction between a flying photon and stationary qubit (electron spin in QD), the initialization (elec-tron spin-superposition state) and manipulation of the spin state can be prepared by optical pumping or optical cooling 78, and can be acquired using pulsed magnetic resonance techniques, nanosecond microwave pulses, or picosecond/femtosecond optical pulses79–82. Therefore, we obtain the interaction of the CRk gate, using the QD-cavity system, with high fidelity for the DQFT scheme. Consequently, according to our analysis results, in practice (under vacuum noise and sideband Figure 6. Fidelities, F1 (QD1 gate: ω − ωc = κ/2) and F2 (QD2 gate: ω − ωc = 0) of the output states for the side- leakage rate κs/κ and coupling strength g/κ with γ/κ = 0.1 and ω ω=−c X under vacuum noise, N(ω), for the operation of the QD-dipole and leaky modes, S(ω) (sideband leakage and absorption). In the tables, the values of fidelities F 1 and F2 are listed for the differences in κs/κ with g/κ = 2.5, and also the differences in g/κ with κs/κ = 1.0. 10 Scientific RepoRtS | (2019) 9:12440 | https://doi.org/10.1038/s41598-019-48695-z www.nature.com/scientificreports www.nature.com/scientificreports/leakage), we can obtain high efficiency and reliable performance of the interaction between a photon and QD within a single-sided optical cavity (CRk gate). Our results also indicate that our DQFT scheme using CRk gates can be experimentally feasible, as well as scalable for multi-qubit DQFT. References 1. Kitaev, A. Quantum measurements and the
Question 99multiple-choice
Quantum simulations of lattice systems often require adaptation of mathematical techniques, especially for non-standard system sizes. Discrete symmetries can play a crucial role in the evolution and observable dynamics of such systems.
In a quantum simulation of a lattice system with L=33 sites, which statement best describes the role of discrete Lorentz symmetry in the observed quantum walk dynamics?
1) It guarantees energy conservation throughout the simulation.
2) It enables efficient encoding of quantum information using only edge sites.
3) It enforces thermal equilibrium by periodically redistributing probabilities.
4) It partitions lattice sites into equivalence classes with identical wavefunction values, leading to collapse and revival patterns.
5) It eliminates the need for numerical decomposition of the Fourier matrix for local implementation.
6) It suppresses all effects of Gaussian noise added to the Hamiltonian.
7) It ensures the tunneling matrix contains only nearest-neighbor terms.
✓ Correct Answer:
The correct answer is 4) It partitions lattice sites into equivalence classes with identical wavefunction values, leading to collapse and revival patterns..
📚 Reference Text:
displayed in Table. I. Its corresponding tunneling ma- trix necessarily involves long-range terms. We apply the QQFT scheme to quantum simulation of this model. Since L=33 is not an integer power of 2, for experimental realization, we perform numerical decomposi- tion of the Fourier matrix38for local implementation of the QQFT. In order to benchmark the performance of our ap- proach, we investigate single-particle quantum walks, which can be measured using quantum microscopes in cold atom ex- periments55,56. We consider quantum walk of a single-particle initialized at site n=0. The time ( t) evolution of its wave- function is denoted as n(t). The spacetime crystallization is described by n(t) at discrete times t=sT, with san inte- ger and Ta time period determined by the Lorentz invariant energy dispersion49. The symmetric properties are captured by the matrixGns n(sT). The discrete Lorentz symmetry implies49, Gns=Gn0s0; (29)with s0 n0! =L s n! (mod L): (30) Here, 2 takes integer values, and the coordinates nand sare defined by modulo L. On the spacetime lattice f(s;n)g with 0s;nL 1, the transformation connects one site to another. The LLsites are then partitioned into a few equiva- lence classes in each of which the wavefunction Gnsmust have the same value. As shown in Fig. 3(a), Gdisplays a periodic collapse and revival pattern, which respects the Lorentz sym- metry (Eq. (29)). It has been established by one of the authors that the nontrivial quantum revival dynamics is a consequence of the Lorentz symmetry49. Considering the Li atom experiment setup described above, the total evolution time to simulate the Poincar ´e dynamics in Fig. 3 with system size L=33 is estimated to be 2 :7 seconds. This in principle can be improved by analytically decompos- ing the QQFT according to 33 =311. B. Robustness of simulations against noise To quantify the robustness against imperfections potentially existent in experiments, we add Gaussian noise to the Hamil- tonian sequence (Eq. (5)), replacing H[s] pby (1 +s)H[s] p, with sa random variable drawn from Gaussian distribution char- acterized by the standard deviation . The consequent e ects on the quantum spatiotemporal dynamics are shown in Fig. 3. With increasing the noise strength, we find that the discrete Poincar ´e symmetry is gradually broken. The symmetric re- vival dynamics is evident even at a noise level of =510 3. We provide a quantitative analysis of the noise-induced Poincar ´e symmetry breaking. If a particle is located at the siten1at the initial time t=0, we then use n1+n(t) to denote its wave function at the time t=sTand the site n1+n. Then, Pn1(s;n)= n1+n(t) 2denotes the probability of a particle hop- ping from the initial site n1to the site n1+nat the time sT. For 7 the conservation of probability,P nPn1(s;n)1 must hold for arbitrary n1ands. The translation symmetry guarantees that Pn1(s;n) is independent of n1. More important, the Lorentz symmetry guarantees P(s;n)=P(s0;n0) for ( s;n) and ( s0;n0) satisfying Eq. (30). The equivalence relation (30) partitions all the lattice sites f(s;n)ginto several equivalence classes, de- noted as C
Question 100multiple-choice
Representation theory of algebraic groups studies modules and their bases using combinatorial and geometric methods. Demazure modules and moment polytopes are closely connected to the structure of Lie algebras and the computational complexity of related problems.
Which of the following statements most accurately describes a key outcome of Lakshmibai's inductive construction for Demazure modules in terms of computational complexity and representation theory?
1) The construction provides a basis for Demazure modules, but requires exponential time due to the complexity of the Bruhat order.
2) The basis for Demazure modules cannot be extended to reducible modules or tensor products.
3) Moment polytope membership is undecidable because monomial bases cannot be computed efficiently.
4) The Bruhat order and reduced decomposition computations must be approximated, as exact computation is infeasible.
5) Lakshmibai monomial bases are only practical for small Weyl groups due to computational constraints.
6) The weight computations for Lakshmibai bases require non-polynomial time algorithms for general Lie algebras.
7) Lakshmibai's method produces explicit monomial bases for Demazure modules, and all steps—including Bruhat order, reduced decompositions, and tensor product decompositions—are computable in polynomial time, enabling efficient algorithms for moment polytope membership.
✓ Correct Answer:
The correct answer is 7) Lakshmibai's method produces explicit monomial bases for Demazure modules, and all steps—including Bruhat order, reduced decompositions, and tensor product decompositions—are computable in polynomial time, enabling efficient algorithms for moment polytope membership..
📚 Reference Text:
Note that M;1=Cv, whileM;w`=M. We will now describe Lakshmibai’s inductive construction of sets M;wrof monomials of the form x=fns sfn1 1, with i2PS(G)and eachni>0, such that each B;wr:=(M;wr)vis a basis of the Demazure module M;wr. It will be useful to define w(x) :=s ss 1and (x) := Ps i=1ni ifor any such monomial. For r= 0, we defineM;w 0:=f1g. ThusB;wr=fvgin this case. For r>0, we consider (8) N;wr 1:=fx2M;wr 1:wr 16s rw(x)g; wheredenotes the Bruhat order, noting that M;wr 1has already been defined by the induction hypothesis. For each x2N;wr 1, lettxdenote the weight of (x)v with respect to sl r 2, which is a positive integer [ 29]. That is, tx= (x)(h r), since (x)is the gC-weight of(x)v. We finally define (9)M;wr:=M;wr 1[ffi rx:x2N;wr 1;i= 1;:::;txg; which is a disjoint union. Lakshmibai has shown that, for each r= 1;:::;`,B;wr= (M;wr)vis a basis of the Demazure module M;wr(see [29, Theorem 4.1] and its proof). In particular, B:=B;w`is a basis of the irreducible representation M=M, indexed by the monomials in M:=M;w`. We callBtheLakshmibai monomial basisofM. From the perspective of computational complexity, the crucial observation is thatBcan be constructed in polynomial time: Lemma10.GivenGandM=Mas specified in section 3.1, the set of monomials Mcan be constructed in polynomial time. Proof.Since the length `=`(w)of the longest Weyl group element is poly(hGi), it suffices to show that, for each r= 1;:::;`,M;wrcan be computed from M;wr 1in time poly(hGi;hMi). We first argue that N;wr 1as defined in (8) can be constructed in polynomial time. For this, we note that both wr 1ands rw(x)for anyx2M;r 1 are given by their reduced decompositions [ 29]. But for any two Weyl group elements w0;w002W, given by their reduced decompositions, it can be decided in poly(hGi) time whether w0w00(ifgis of classical type, this result can be deduced from [ 37, Theorems 5A, 5BC, 5D]; the five exceptional Lie algebras can be treated separately in constant time). Since #M;r 1=dimM;wr 1dimMhMi, it is now easy to see thatN;wr 1can be constructed in time poly(hGi;hMi). After this,M;rcan be constructed via (9) likewise in poly(hGi;hMi). We now establish Lemmas 8 and 9: Proof of Lemma 8.Ifgis the compact real form of simple Lie algebra and Mis irreducible then this follows directly from Lemma 10: First compute Mand then add MEMBERSHIP IN MOMENT POLYTOPES IS IN NP AND CONP 13 the weight (x)of(x)vfor allx2Minto the multiset. If gis one-dimensional abelian and Mirreducible then there is only a single weight, which we already know from the specification of M. Ifg=g1 gnis a direct sum of such Lie algebras and Mirreducible, then any irreducible representation is a tensor product of irreducible gi-representations for i= 1;:::;n, and the multiset of weights can be identified with the Cartesian product of the multiset of weights of its constituents, which can be computed in polynomial time. Finally, if Mis reducible we apply the above procedure to each irreducible summand in its specification. Proof of Lemma 9.We may likewise assume that gCis simple and Mis irreducible, i.e.,M=Mfor some highest weight . It moreover suffices to show that the representation matrices of e ,f =e , andh for
Question 101multiple-choice
Quantum phase estimation is a fundamental algorithm in quantum computing, used to estimate the eigenvalues of unitary operators with high precision. Managing errors, coherence, and auxiliary states are key considerations in the practical implementation and efficiency of such algorithms.
Which statement most accurately describes the optimal runtime scaling for quantum phase estimation algorithms in terms of the promise gap α and error tolerance δ?
1) The runtime scales as O(α log(δ⁻¹)), improving as α increases.
2) The runtime is independent of both α and δ for sufficiently large systems.
3) The runtime scales quadratically with respect to α⁻¹ and logarithmically with δ⁻¹.
4) The runtime is determined only by the number of qubits, not by α or δ.
5) The runtime has an exponential dependence on δ⁻¹ and no dependence on α.
6) The runtime scales as O(α⁻¹ + log(δ⁻¹)), combining linear and logarithmic terms.
7) The runtime scales as O(2ⁿα⁻¹log(δ⁻¹)), and the α⁻¹ dependence is shown to be optimal based on approximate counting lower bounds.
✓ Correct Answer:
The correct answer is 7) The runtime scales as O(2ⁿα⁻¹log(δ⁻¹)), and the α⁻¹ dependence is shown to be optimal based on approximate counting lower bounds..
📚 Reference Text:
to make the ampli- tudes always close to 0 or 1. We claim that all of these attempts fail, and furthermore that achieving coherent phase estimation without some kind of promise is impossible in principle. This is due to the polynomial method argument above: any coherent quantum algorithm’s output state’s amplitudes must be a continuous function of the phase, and continuous functions that are sometimes ≈0and sometimes≈1must somewhere have an intermediate value. Recall that uncomputation only works when the amplitudes are close to 0 or 1. The error depends on if the phase is close to this transition point or not, so the error must depend on the phase. This issue was not taken into account by [Ambainis10], [KP17], and [KLLP18], since all of these either implicitly or explicitly state that there is an algorithm that approximately performs |ψi/angbracketright|0n/angbracketright→|ψi/angbracketright|λi/angbracketrightin superposition while λi Accepted in Quantum2021-10-14, click title to verify. Published under CC-BY 4.0. 6 is a deterministic computational basis state1. A more promising approach is detailed in [TaShma13], which, crudely speaking, shifts the transition points by a classically chosen random amount. Now whether or not a phase is close to a transition point is independent of the phase itself, making amplification possible. [Ambainis10] describes a similar idea, calling it ‘unique-answer’ eigenvalue estimation. However, we claim that this only works for a single phase. If we consider, for example, a unitary whose phases are uniformly dis- tributed in [0,1)at a sufficiently high density, then there will be a phase near a transition point for any choice of random shift. The only way to avoid the rounding promise is to sacrifice coherence and measure the output state. All of the algorithms in this paper, including textbook phase estimation, achieve an asymptotic runtime of O(2nα−1log(δ−1)), whereδis the error in diamond norm. Before we move on to the formal definition of the estimation task, we informally argue that the α−1dependence is optimal, via a reduction to approximate counting. We are given N items,Kof which are marked. Following the standard method for approximate counting [BHMT00], we construct a Grover unitary whose phases λjencode arcsin(/radicalbig K/N ). Given a(1,α)-rounding promise, computing floor (21λj)amounts to deciding if λj≤1−α/2 2or λj≥1+α/2 2given that one of these is the case. By shifting the λjaround appropriately we can thus decide if K≥(1/2 +Cα)NorK≤(1/2−Cα)N, for some constant Cobtained by linearising arcsin(/radicalbig K/N ). We have achieved approximate counting with a promise gap ∼α. Thus the Ω(α−1)lower bound on approximate counting [NW98] implies our runtime must be Ω(α−1). Equipped with the notion of a rounding promise, we can define our estimation tasks. Many algorithms in this paper produce some kind of garbage, which can be dealt with the uncompute trick [BBBV97]. Rather than repeat the analysis of uncomputation in every single proof, we present a modular framework where we can deal with uncomputation separately. Furthermore, some applications may require computing some function of the final estimate, resulting in more garbage which also needs to be uncomputed. Rather than baking the uncomputation into
Question 102multiple-choice
Isogeny-based cryptography leverages the mathematical structure of elliptic curves and their isogenies to construct protocols aimed at resisting attacks from both classical and quantum adversaries. Efficient and secure representations of isogenies are critical for the practicality and safety of such protocols.
Which of the following statements best explains why suborder representations in isogeny-based cryptography fail to provide quantum resistance, even for isogenies of prime degree?
1) Quantum algorithms solving the hidden subgroup problem can recover the endomorphism ring from suborder representations, undermining security.
2) Suborder representations inherently disclose the full isogeny kernel, allowing direct computation of secret keys.
3) The use of suborder representations increases the degree of the isogeny, making brute-force search feasible for quantum computers.
4) Suborder representations rely on polynomials whose coefficients can be efficiently factored with quantum algorithms.
5) Even when using suborder representations, knowledge of only the domain curve is sufficient to reconstruct the shared secret.
6) Suborder representations are equivalent to ideal representations, which are not vulnerable to quantum attacks.
7) The suborder representation only protects against classical attacks and does not consider the algebraic structure exposed to quantum adversaries.
✓ Correct Answer:
The correct answer is 1) Quantum algorithms solving the hidden subgroup problem can recover the endomorphism ring from suborder representations, undermining security..
📚 Reference Text:
setting of pSIDH [ Ler22a]). Indeed, in that case, the standard ways to represent isogenies(with polynomials, or kernel points) are not compact or efficient enough to match our definition. The Deuring correspondence gave us the tools to obtain efficient re presenta- tions with a natural isogeny representation obtained by taking sφas the ideal Iφ corresponding to φ. This ideal representation matches Definition 2.2, however it 8 M. Chen, M. Imran, G.Ivanyos, P.Kutas, A.Leroux, C. Petit also reveals the endomorphism ring of E′. One of the motivations of Leroux in [Ler22a] to introduce another isogeny representation (called the subord er rep- resentation) is to have an isogeny representation that does not d irectly reveal the endomorphism ring of the codomain. This suborder representa tion matches our notion of weak isogeny representation as defined in Definition 2.2 . The main contribution of this paper implies that the suborder representatio n does not hide the endomorphism ring of the codomain to a quantum computer, eve n when the degree is prime. Since then, Robert [ Rob22] suggested to use the techniques introduced to attack SIDH in order to obtain another isogeny representation (t his one not even requiring to reveal the endomorphism ring of the domain). Our analysis holds for any suborder representation, hence it also applies to Rob ert’s one. 2.5 The pSIDH key exchange As an application of the hardness of computing the endomorphism rin g from the suborder representation, Leroux introduced a key exchang e called pSIDH. The principle can be summarized as follows: use the evaluation algorith m for the suborder representation to perform an SIDH-like key exchange, but for isogenies of big prime degree. The SIDH and pSIDH key exchange both use the following commutative isogeny diagram: EBψA/d47/d47EAB E0φB/d79/d79 φA/d47/d47EAψB/d79/d79 In pSIDH, Alice and Bob’s secret keys are ideal representations fo r the iso- geniesφAandφB(or equivalently the endomorphism ring of the two curves EA andEB), and their associated public keys are the suborder representat ions for φAandφB. Leroux showed that the knowledge of End( EA) (resp. End( EB)) and the suborder representation of φB(resp.φA) was enough to compute the end curve EABfrom which the common secret can be derived efficiently. The mechan ism behind this computation is quite complicated and is not relevant for us since we target the key recovery problem. We refer to [ Ler22a] for more details. 2.6 The hidden subgroup problem The hidden subgroup problem (HSP for short) in a group Gis defined as the problemoffinding a subgroup H≤Ggivena function fonGsatisfyingthat fis constant on the left cosets of Hand takes different values on different cosets, i.e., f(x) =f(y) if and only if x−1y∈H. There is also a right version of the hidden subgroupproblemwherethelevelsetsofthehidingfunction faretherightcosets ofH. As taking inverses in Gmaps left cosets to right cosets and vice versa, Title Suppressed Due to Excessive Length 9 the two versions of HSP are equivalent. (One just needs to replace the hiding function with its composition with taking inverses.) Although the equiv alence is straightforward, it is
Question 103multiple-choice
Integrable systems often utilize matrix identities and τ-functions to describe their solutions, with connections extending to quantum groups and representation theory. The transition from classical to quantum cases involves introducing deformation parameters and new algebraic structures.
Which construction is necessary when extending τ-functions from the classical group SL(n) to its quantum counterpart SLq(n) for q ≠ 1?
1) Replacing Schur polynomials with Legendre polynomials
2) Introducing time variables as complex conjugates
3) Utilizing ordinary determinants for all calculations
4) Imposing the trace condition instead of the determinant condition
5) Switching from difference to differential equations
6) Enforcing commutativity of all group elements
7) Introducing q-antisymmetrization and q-determinants
✓ Correct Answer:
The correct answer is 7) Introducing q-antisymmetrization and q-determinants.
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[β1...βk/parenrightigg H/parenleftiggα′ kα′ 1...α′ k−1 βk+1]β′ 1...β′ k−1/parenrightigg =H/parenleftiggα1...αk[α′ k β1...βkβk+1/parenrightigg H/parenleftiggα′ 1...αk−1]′ β′ 1...β′ k−1/parenrightigg .(77) Just like original (67) these are just matrix identities, valid for any Hα β. However, after the switch from gtoHwe, first, essentially represented the equations in n-independent form and, second, opened the possibility to rewrite them in terms of time-deriv atives. 21 For example, in the simplest case of αi=i, i= 1,...,k′; βi=i, i= 1,...,k+1; α′ i=i, i= 1,...,k−1, α′ k=k+1; β′ i=i, i= 1,...,k−1(78) we get: H/parenleftigg1...k 1...k/parenrightigg H/parenleftiggk+1,1...k−1 k+1,1...k−1/parenrightigg − −H/parenleftigg1...k−1,k 1...k−1,k+1/parenrightigg H/parenleftiggk+1,1...k−1 k,1...,k−1/parenrightigg = =H/parenleftigg1...k+1 1...k+1/parenrightigg H/parenleftigg1...k−1 1...k−1/parenrightigg(79) (all other terms arising in the process of symmetrization vanish). T his in turn can be repre- sented through τ-functions: ∂1¯∂1τ(k)·τ(k)−¯∂1τ(k)∂τ(k)=τ(k+1)τ(k−1). (80) This is the usual lowest Toda-lattice equation. For finite nthe set of solutions is labeled by g∈SL(n) as a result of the additional constraints (75). We can now use the chance to illustrate the ambiguity of definition of τ-function, or, to put it differently, that in the choice of time-variables. Eq.(80) is actu ally a corollary of two statements: the basic identity (67) and the particular definition (1 ), which in this case implies (74) withP’s being ordinary Schur polynomials (71). At least, in this simple situatio n (of fundamental representations of SL(n)) one could define τ-function not by eq.(1), but just by eq.(73), with Hα β(s,¯s)−→ Hα β(s,¯s) =/summationdisplay i,jPi−α(s)gi jPj−β(¯s) (81) withanyset of independent functions (not even polynomials) Pα. Such τ(k) P= det 1≤α,β≤kHα β (82) 22 still remains a generating function for all matrix elements of G=SL(n) in representation F(k). This freedom should be kept in mind when dealing with “generalized τ-functions”. As a simple example, one can take Pα(s) to beq-Schur polynomials, /productdisplay ieq(sizi) =/summationdisplay jP(q) j(s)zj,or /productdisplay ieqi(sizi) =/summationdisplay jˆP(q) j(s)zj,(83) which satisfy (hereafter we denote D≡D(0)) DsiP(q) j(s) = (Ds1)iP(q) j(s) =P(q) j−i(s). (84) Then instead of (76) we would have: DsiHα β=Hα+i β, D¯siHα β=Hα β+i(85) and τ(k) P(q)(s,¯s|g) = det 1≤α,β≤kDα−1 s1Dβ−1 ¯s1H1 1(s,¯s). (86) So definedτ-function satisfies difference rather than differential equations [14, 15]: τ(k)·Ds1D¯s1τ(k)−Ds1τ(k)·D¯s1τ(k)=τ(k−1)·M+ s1M+ ¯s1τ(k+1), ... .(87) We emphasize, however, that, in a sense, this is just a redefinition o f theSL(n), notSLq(n) τ-function, as generating function of matrix elements. In particula r, thisτ-function is a c- rather than q-number function. Still it would have something to do with SLq(n) group, but as a function of times, i.e. rather in spirit of the connection of q-hypergeometric functions to quantum groups (see, for example, [16]). 5.2 Approach to SLq(n) In order to extend this reasoning to the case of SLq(n) withq/ne}ationslash= 1 we need to go into some more details about the group structure. 23 The main thing we shall need is the notion of q-antisymmetrization, to be defined as a sum over all perturbations, ([1,...,k]q) =/summationdisplay P(−q)degP(P(1),...,P(k)), (88) where degP= # of inversions in P. (89) The first place to use this notion is the definition of q-determinant: detqA∼A[1 [1...An]q n]q=/summationdisplay P,P′(−q)degP+degP′/productdisplay aAP(a) P′(a). (90) Note that this is
Question 104multiple-choice
In distributed quantum computing, efficient circuit compilation is essential for large-scale computations, especially when using hardware architectures that limit qubit interactions to nearest neighbors. Techniques that exploit circuit topology and strategic qubit placement can significantly improve performance and scalability.
Which approach most effectively reduces the number of SWAP gates required in quantum circuits with Linear Nearest Neighbor (LNN) architecture when implementing algorithms characterized by high local connectivity and sparse full connectivity?
1) Increasing the number of non-local two-qubit gates
2) Utilizing solely fully connected qubit layouts
3) Leveraging dangling qubits to facilitate interactions
4) Applying only single-qubit gates throughout the circuit
5) Minimizing the use of ancillary qubits
6) Employing random qubit assignments
7) Restricting the quantum algorithm to classical subroutines
✓ Correct Answer:
The correct answer is 3) Leveraging dangling qubits to facilitate interactions.
📚 Reference Text:
Title: ECDQC: Efficient Compilation for Distributed Quantum Computing with Linear Layout Year: 2024 Paper ID: 8f613e3d9f5d0d3e19190ae5a99cacb8eb2a29cb Source: semantic-scholar URL: https://www.semanticscholar.org/paper/8f613e3d9f5d0d3e19190ae5a99cacb8eb2a29cb Abstract: In this paper, we propose an efficient compilation method for distributed quantum computing (DQC) using the Linear Nearest Neighbor (LNN) architecture. By exploiting the LNN topology’s symmetry, we optimize quantum circuit compilation for High Local Connectivity, Sparse Full Connectivity (HLC-SFC) algorithms like Quantum Approximate Optimization Algorithm (QAOA) and Quantum Fourier Transform (QFT). We also utilize dangling qubits to minimize non-local interactions and reduce SWAP gates. Our approach significantly decreases compilation time, gate count, and circuit depth, improving scalability and robustness for large-scale quantum computations.
Question 105multiple-choice
In supersymmetric theories inspired by string models, the properties of moduli fields and their interactions with the visible sector have significant implications for cosmology and particle physics, particularly regarding dark matter and collider searches. The mass spectrum of superpartners is closely tied to the mechanism by which supersymmetry breaking is mediated.
Which supersymmetry breaking mediation scenario typically allows gaugino superpartners to be light enough for detection at the LHC, while requiring suppression of gravity-mediated scalar masses to realize this spectrum?
1) Gravity mediation with unsuppressed scalar masses
2) Gauge mediation with high messenger scales
3) Pure gravity mediation with Planck-scale couplings
4) Mini-split supersymmetry with heavy scalars and gauginos
5) Anomaly mediation (AMSB) or AMSB-like scenarios with suppressed gravity-mediated contributions
6) Gauge mediation with extremely light gravitinos
7) Pure moduli mediation without anomaly or gauge contributions
✓ Correct Answer:
The correct answer is 5) Anomaly mediation (AMSB) or AMSB-like scenarios with suppressed gravity-mediated contributions.
📚 Reference Text:
moduli are not completely understood, certain features do seem to be fairly universal. For example, moduli masses of m'm3=2are expected when the potential arises mainly from supersymmetry breaking [18]. Moduli may also have supersymmetric potentials [19], and m' m3=2is found in some cases [20, 21]. However, m'm3=2is still frequently obtained from supersymmetric potentials once the constraint of a very small vacuum energy is imposed [21]. Thus, a plausible generic expectation from string theory is that there exists at least one modulus eld with m'm3=2and MPl[6].2Other heavier moduli may be present, but since the lightest and most weakly-coupled modulus will decay the latest, it is expected to have the greatest impact on the present-day cosmology. Putting these two pieces together, acceptable reheating from string moduli suggests m' m3=2&100 TeV. This has important implications for the masses of the SM superpartner elds. Surveying the most popular mechanisms of supersymmetry breaking mediation [24], the typical size of the superpartner masses is msoft8 >< >:m3=2 gravity mediation LMPl M m3=2gauge mediation Lm 3=2 anomaly mediation(2) 1Such an early matter-dominated phase might also leave an observable signal in gravitational waves at multiple frequencies [11] or modify cosmological observables [12{15]. 2The LARGE Volume Scenario of Refs. [22, 23] is a notable exception to this. 3 whereLg2=(4)2is a typical loop factor and M MPl=Lis the mass of the gauge messengers. Of these mechanisms, only anomaly mediation (AMSB) allows for superpartners that are light enough to be directly observable at the Large Hadron Collider (LHC) [25, 26]. Contributions to the soft terms of similar size can also be generated by the moduli themselves [27, 28], or other sources [29{32]. However, for these AMSB and AMSB-like contributions to be dominant, the gravity-mediated contributions must be suppressed [25], which is non-trivial for the scalar soft masses [33{37]. An interesting intermediate scenario is mini-split supersymmetry where the dominant scalar soft masses come from direct gravity mediation with msoftm3=2, while the gaugino soft masses are AMSB-like [38{43]. Moduli reheating can also modify dark matter production [16, 44{46, 48]. A standard weakly-interacting massive particle (WIMP) will undergo thermal freeze-out at temper- atures near Tfom =20. If this is larger than the reheating temperature, the WIMP density will be strongly diluted by the entropy generated from moduli decays. On the other hand, DM can be created non-thermally as moduli decay products. A compelling picture of non-thermal dark matter arises very naturally for string-like moduli and an AMSB-like superpartner mass spectrum [16]. The lightest (viable) superpartner (LSP) in this case tends to be a wino-like neutralino. These annihilate too eciently to give the observed relic density through thermal freeze-out [49{51]. However, with moduli domination and reheating, the wino LSP can be created non-thermally in moduli decays, and the correct DM density is obtained for M2200 GeV and m'3000 TeV. This scenario works precisely because the wino annihilation cross section is larger than what is needed for thermal freeze-out. Unfortunately, such enhanced annihilation rates are strongly constrained by gamma-ray observations of the galactic centre by Fermi
Question 106multiple-choice
Quantum algorithms leverage the principles of quantum mechanics to solve certain computational problems more efficiently than classical algorithms. Various paradigms exist, including hidden subgroup approaches, quantum walks, and topological methods, each offering distinct advantages for specific problem classes.
Which quantum algorithm framework provides exponential speedup for integer factoring and discrete logarithm problems by exploiting Abelian group structures?
1) Quantum walk algorithms
2) Adiabatic algorithms
3) Abelian Hidden Subgroup algorithms
4) Topological quantum algorithms
5) Quantum simulation algorithms
6) Grover's amplitude amplification
7) Non-Abelian Hidden Subgroup algorithms
✓ Correct Answer:
The correct answer is 3) Abelian Hidden Subgroup algorithms.
📚 Reference Text:
Title: Quantum Algorithms Year: 2008 Paper ID: d6fc3ebbb9d4dc301279bd2979c7826b60d9d395 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/d6fc3ebbb9d4dc301279bd2979c7826b60d9d395 Abstract: This article surveys the state of the art in quantum computer algorithms, including both black-box and non-black-box results. It is infeasible to detail all the known quantum algorithms, so a representative sample is given. This includes a summary of the early quantum algorithms, a description of the Abelian Hidden Subgroup algorithms (including Shor's factoring and discrete logarithm algorithms), quantum searching and amplitude amplification, quantum algorithms for simulating quantum mechanical systems, several non-trivial generalizations of the Abelian Hidden Subgroup Problem (and related techniques), the quantum walk paradigm for quantum algorithms, the paradigm of adiabatic algorithms, a family of ``topological'' algorithms, and algorithms for quantum tasks which cannot be done by a classical computer, followed by a discussion.
Question 107multiple-choice
In the theory of quantum groups and representation theory, the universal fusion matrix plays a key role in describing how representations can be fused while respecting underlying algebraic structures. The ABRR equation uniquely determines this fusion matrix through a functional relation involving certain operators.
Which property guarantees the existence of a unique solution to the ABRR equation for the universal fusion matrix J(λ) in both quantum group and Lie algebra settings?
1) The commutativity of the Cartan subalgebra elements
2) The invertibility of the R-matrix
3) The semisimplicity of the underlying algebra
4) The centrality of the Casimir operator
5) The surjectivity of the groupoid morphism
6) The recursive construction based on the structure of the root system
7) The symmetry of the Poisson bracket
✓ Correct Answer:
The correct answer is 6) The recursive construction based on the structure of the root system.
📚 Reference Text:
which equips X with the structure of a Poisson groupoid with base U. Moreover, it is easy to check that the Poisson groupoid Xis naturally isomorphic to Xr. 8 Theuniversal fusion matrix and theArnaudon- Buffenoir-Ragoucy-Roche equation 8.1. The ABRR equation. In [ABRR], Arnaudon, Buffenoir, Ragoucy and Roche give a general method for constructing the universal fusio n matrix J(λ), which lives in some completion of Uq(g)⊗2, i.e the unique element satisfying JV W(λ) =J(λ)|V⊗Wfor allV,W. A similar approach is suggested in [JKOS], based on the method of [Fr] LetU′(b±) be the kernel of the projection U(b±)→U(h). We use the same notations with the index qfor the quantum analogs of these objects. We setθ(λ) =λ+ρ−1 2/summationtext ix2 i∈Uhwhere as usual ρ=1 2/summationtext α∈∆+hαand (xi) is an orthonormal basis of h. SetR0=Rq−/summationtextxi⊗xi. It is known that R0∈1+U′ q(b+)⊗U′ q(b−). Theorem 8.1 ([ABRR]). The universal fusion matrix J(λ)ofUq(g)is the unique solution of the form 1+U′ q(b−)⊗U′ q(b+)of the equation J(λ)(1⊗q2θ(λ)) =R21 0(1⊗q2θ(λ))J(λ). (8.1) The universal fusion matrix J(λ)ofU(g)is the unique solution of the form 1+U′(b−)⊗U′(b+)of the equation [J(λ),1⊗θ(λ)] = (/summationdisplay α∈∆+e−α⊗eα)J(λ) (8.2) We will call these equations the ABRR equations forUq(g) and for g, re- spectively. Proof.Let us first show the statement about uniqueness. Let T(λ)∈1 + U′ q(b−)⊗U′ q(b+) be any solution of (8.1). Then (R21 0)−1T(λ) = Ad (1 ⊗q2θ(λ))T(λ) ⇔((R21 0)−1−1)T(λ) = (Ad (1 ⊗q2θ(λ))−1)T(λ) ⇔T(λ) = 1+(Ad (1 ⊗q2θ(λ))−1)−1((R21 0)−1−1)T(λ) Now notice that (( R21 0)−1−1)∈U′ q(b−)⊗U′ q(b+). This implies that T(λ) can be recusively constructed as follows. Set T0(λ) = 1 and put Tn+1(λ) = 1+(Ad (1 ⊗q2θ(λ))−1)−1((R21 0)−1−1)Tn(λ). 23 Then lim n→∞Tn(λ) =T(λ) (the limit is in the sense of stabilization). In par- ticular there exists a unique solution to (8.1) of the given form. The proof in the rational case (i.e in the case of a simple Lie algebra g) is similar. In that case, the recursive construction is given by T0(λ) = 1 and Tn+1(λ) = 1−ad(1⊗θ(λ)−1)(/summationdisplay α∈∆+e−α⊗eα)Tn(λ). We now give a proof that the fusion matrix J(λ) actually satisfies the ABRR relation in the case of simple Lie algebras. The proof in the case of qua ntum groups is analogous but technically more challenging, and is given in App endix B. LetCbe the quadratic Casimir operator in the center of the universal en - veloping algebra Ug: C=/summationdisplay ix2 i+2ρ+2/summationdisplay α∈∆+e−αeα. ThenCacts on any highest weight representation of gof highest weight λby the scalar ( λ,λ+ 2ρ). Now let V,Wbe two finite-dimensional g-modules and letv∈V, w∈Wbe two homogeneous elements of weight wt( v) and wt( w). We compute the quantity F(λ) =/a\}bracketle{tv∗ λ−wt(v)−wt(w),Φw λ−wt(v)(C⊗1)Φv λvλ/a\}bracketri}ht in two different ways. On one hand we have F= (λ−wt(v),λ−wt(v)+2ρ)J(λ)(w⊗v). (8.3) On the other hand, F=/a\}bracketle{tv∗ λ−wt(v)−wt(w),/braceleftbig 2((e−αeα)1+(e−αeα)2+(eα⊗e−α)12+(e−α⊗eα)12+ρ1+ρ2) +/summationdisplay i(x2 i)1+(x2 i)2+2(xi⊗xi)12/bracerightbig Φw λ−wt(v)Φv λvλ/a\}bracketri}ht. Sincev∗ λ−wt(v)−wt(w)is a highest weight vector, it is clear that /a\}bracketle{tv∗ λ−wt(v)−wt(w),(e−αeα)1Φw λ−wt(v)Φv λvλ/a\}bracketri}ht=/a\}bracketle{tv∗ λ−wt(v)−wt(w),(e−α⊗eα)12Φw λ−wt(v)Φv λvλ/a\}bracketri}ht= 0. Moreover, by the intertwining property again, we have (eα⊗e−α)12Φw λ−wt(v)Φv λvλ=−(e−αeα)2−(e−α⊗eα)23Φw λ−wt(v)Φv λvλ, (ρ1+ρ2)Φw λ−wt(v)Φv λvλ=−ρ3Φw λ−wt(v)Φv λvλ+Φw λ−wt(v)Φv λρvλ, ((x2 i)1+(x2 i)2+2(xi⊗xi)12)Φw λ−wt(v)Φv λvλ=−(2(xi⊗xi)13+2(xi⊗xi)23+(x2
Question 108multiple-choice
Quantum computing leverages probabilistic phenomena to solve certain computational problems more efficiently than classical approaches. One notable application is integer factorization, which is fundamental to modern cryptography.
Which of the following best describes how the Chernoff bound is utilized in quantum algorithms to reduce error probability when using majority voting across multiple runs?
1) It guarantees that the algorithm will always produce the correct result after a finite number of repetitions.
2) It provides a linear decrease in failure probability as the number of runs increases.
3) It determines the minimum number of runs required to achieve a 100% success rate.
4) It ensures that hardware errors are eliminated during repeated executions.
5) It allows the algorithm to deterministically amplify quantum interference effects during computation.
6) It bounds the probability of algorithmic failure by showing that, after k independent runs, the chance of majority error decreases exponentially with k.
7) It replaces the need for majority voting by enabling a single-run quantum algorithm to achieve arbitrary precision.
✓ Correct Answer:
The correct answer is 6) It bounds the probability of algorithmic failure by showing that, after k independent runs, the chance of majority error decreases exponentially with k..
📚 Reference Text:
probabilistic devices and can be viewed as a generalization of a probabilistic Turing machine. It is hardly surprising that quantum algorithms exhibit their probabilistic nature by producing a solution with some probabil- ity. Suppose we have an algorithm which succeeds with probability1 2+dfor some fixed constant d. Then the probability of failure for this algorithm is1 2 d. 21 The algorithm is run ktimes and we take the majority result. Then the probability that the algorithm fails is given by Pr=bk 2c å l=0k l1 2+dl1 2 dk l (2.62) where lis the number of times the algorithm returns the correct answer. Forl=k 2 we have 1 2+dbk 2c1 2 dk bk 2c =1 2+dbk 2c1 2 ddk 2e =1+2d 2bk 2c1 2d 2dk 2e (2.63) 1 4d2 4k 2 = 1 4d2k 2 2k: Since Eq (2.62) contains at most 2kterms l=k 2 , the failure probability is bounded as follows: Pr2k2 4 1 4d2k 2 2k3 5= 1 4d2k 2: (2.64) Application of 1 xe x(2.65) produces 1 4d2k 2 e 4d2k 2=e 2d2k(2.66) and we obtain a Chernoff bound Pre 2d2k: (2.67) Therefore, the algorithm succeeds with probability exponentially close to 1 in the number of trials. 22 To illustrate the point convincingly, suppose the algorithms fails with probability1 4, i.e., d=1 4. The approximate Chernoff bounds corresponding to an increasing number of algorithm repetitions are given in Table 2.2. Table 2.2: The Chernoff bounds for d=1 4 k e 2d2k 10 0:286 40 0:007 100 3:7310 6 400 1:9310 22 1000 5:1710 55 When the probability drops below 10 20, a failure of the computer itself is more likely than that of an algorithm. For 1000 runs, you are more likely to be hit by an asteroid while reading this sentence than to get a wrong answer from the algorithm! 23 CHAPTER 3 SHOR’S FACTORIZATION ALGORITHM The existence of a unique decomposition of any integer N>1 into a product of primes was formalized by Euclid around 300 BC in Propositions 30 and 32 of his Elements [18]. The first complete proof was provided in 1801 by Carl Friedrich Gauss in his Disquisitiones Arithmeticae , where he remarked [19]: The problem of distinguishing prime numbers from composite numbers, and of re- solving the latter into their prime factors is known to be one of the most important and useful in arithmetic. The dignity of the science itself seems to require that every possible means be explored for the solution of a problem so elegant and so celebrated. Almost two centuries later Peter Shor discovered a quantum algorithm that factors an integer N with high probability in O n2log2nlog2log2n (3.1) steps where nis the number of bits needed to represent N, an exponential speedup over the best- known classical algorithm. 3.1 Reduction of Factorization to Order Finding We may assume that an integer Nwe wish to factor is odd, as factors of 2 can be easily removed. Moreover, since there exist efficient classical tests for prime powers [30], we assume thatNis an odd non-prime power integer. Miller [42] showed that factorization reduces to order finding. 24 Choose a random
Question 109multiple-choice
In quantum information theory, algebraic structures such as the Temperley-Lieb Algebra (TLA) and braid group representations play a central role in understanding and engineering entanglement in multi-qubit systems. The action of specific unitary operators constructed from tensor products of Hermitian matrices determines how quantum states are transformed and can lead to various degrees of entanglement.
Which choice describes a situation in which the action of a unitary braiding operator on an n-qubit separable state results in the state remaining fully separable rather than entangled?
1) All Hermitian operators \( s_j \) are chosen as the Pauli \( \sigma_1 \) matrix in the tensor product construction
2) All Hermitian operators \( s_j \) are the identity matrix
3) All Hermitian operators \( s_j \) are the Hadamard matrix
4) Some \( s_j \) are Pauli matrices and others are identity
5) The parameter \( \phi \) is set to \( \pi \) in the construction
6) The normalization condition is not satisfied
7) The operators are chosen from a Fibonacci fusion category
✓ Correct Answer:
The correct answer is 2) All Hermitian operators \( s_j \) are the identity matrix.
📚 Reference Text:
to generalize the above representa- tion of TLA to higher dimensions is as follows. Let e1=/parenleftbigg 1 0 0 0/parenrightbigg , e2=/parenleftbigg a20 0b2/parenrightbigg , e3=/parenleftbigg 0e−iφ eiφ0/parenrightbigg .(11) Define E(n,k) 1≡ ⊗k−1 j=1I⊗e1⊗n j=k+1I, (12) E(n,k) 2≡ ⊗k−1 j=1I⊗e2⊗n j=k+1I +ab⊗k−1 j=1sj⊗e3⊗n j=k+1sj,(13) where⊗m j=1sj=s1⊗s2⊗···⊗sm. Heresjis any Her- mitian operator satisfying s2 j= 1. For example, sjcan beI, any one of the Pauli matrices σm(m= 1,2,3), or the Hadamard matrix H. The integer nis the number of 2×2 matrices in the tensor products, and kindicates the position of e1, e2ande3. TheE(n,k) i’s are 2n×2nma- trices, and they reduce to (9) in the case n=k= 1. One can easily check that E(n,k) i’s satisfy (10). Hence, the op- eratorsh(n,k) i=dE(n,k) iform a 2n×2nmatrix realization ofTL3(d)4. A unitary braid group representation is then obtained from the hi’s by the Jones representation. Our new unitary braid representation generalizes the 2×2 matrices of (9) to 2n×2nmatrices of (13) within the TLATL3(d). Other routes of generalization are pos- sible. For instance, in [14] the 2 ×2 representation of TL3(d) were generalized to higher dimensional matrices forTLm(d) withm >3, where the dimension of repre- sentation varies with the number of strands maccording to the Fibonacci numbers, or with the number of inde- pendent bit-strings of certain path model proposed in [15]. Generalized GHZ states. – From now on we will be mainly concerned with the unitary braiding transfor- mation representing the action of the braid b1b2. This braiding operator can be evaluated to be b(n,k) 1b(n,k) 2 =⊗k−1 j=1I⊗/parenleftbigg da20 0db2+A−2/parenrightbigg ⊗n j=k+1I(14) 4See [9] for an n2×n2matrix realization of the TLA. The braiding operator (called the Yang-Baxter matrix in these works) was obtained there through aYang-Baxterization process. This latter process was also employed in [8], but not related to TLA, to generalize the Bell matrix to (2 n)2×(2n)2braid matrices.+⊗k−1 j=1sj⊗/parenleftbigg 0−e−iφA4dab eiφdab 0/parenrightbigg ⊗n j=k+1sj. Its action on the separable n-qubit states |a1a2...ak−10ak+1···an/an}bracketri}htand|a1a2...ak−11ak+1···an/an}bracketri}ht (aj= 0,1, j= 1,2,...,k−1,k+1,...n) is given by b(n,k) 1b(n,k) 2|a1a2...ak−10ak+1···an/an}bracketri}ht = (da2)|a1a2...ak−10ak+1···an/an}bracketri}ht + (eiφdab)|˜a1˜a2...˜ak−11˜ak+1···˜an/an}bracketri}ht,(15) and b(n,k) 1b(n,k) 2|a1a2...ak−11ak+1···an/an}bracketri}ht = (db2+A−2)|a1a2...ak−11ak+1···an/an}bracketri}ht + (−e−iφA4dab)|˜a1˜a2...˜ak−10˜ak+1···˜an/an}bracketri}ht,(16) where|˜aj/an}bracketri}ht ≡sj|aj/an}bracketri}ht(j= 1,...,k−1,k+1,...,n). Thus under the action of b(n,k) 1b(n,k) 2, the separable n-qubit state|a1a2...ak···an/an}bracketri}htis superimposed on the state |˜a1˜a2...˜ak···˜an/an}bracketri}htineithertheform(15)or(16), depend- ingonwhetherthe k-thqubit |ak/an}bracketri}htis|0/an}bracketri}htor|1/an}bracketri}ht. The states in (15) and (16) arenormalized, as ( da2)2+|eiφdab|2= 1, and|db2+A−2|2+|−e−iφA4dab|2= 1,which can be eas- ily checked. Depending on the choice of the set of sj’s, the resulting state (15) or (16) will have varying degrees of entanglement. In particular, if all sj=I, then the resulting state is separable, and b(n,k) 1b(n,k) 2is simply a local unitary transformation. We now consider a subclass of the representation ob- tained by setting φ= 0 in (13) (i.e., e3=σ1),sj=Ifor j < k, andsj=σ1forj > k. In this case, |˜aj/an}bracketri}ht=|aj/an}bracketri}ht forj k. Hence, un- der the action of B(n,k)≡b(n,k) 1b(n,k) 2(with the above- mentioned choice of the sj’s inb(n,k) iunderstood), the separablen-qubit state |a1a2...ak−1akak+1···an/an}bracketri}htis su- perimposed on the state |a1a2...ak−1¯ak¯ak+1···¯an/an}bracketri}htin either the form (15) or (16) (with the appropriate change in the ˜aj), depending on whether the
Question 110multiple-choice
Computational invariant theory deals with the complexity of algorithms that analyze group actions on vector spaces, often distinguishing between commutative and non-commutative cases. Representation theory provides analytical tools for these problems, particularly when handling non-commutative group actions.
Which feature is essential for guaranteeing polynomial-time performance in algorithms for non-commutative group actions, particularly when using highest weight vectors as potential functions?
1) Allowing arbitrary bit-length group elements in each iteration
2) Ignoring the difference between uniform and non-uniform cases
3) Relying solely on black-box applications of previous results without explicit constructions
4) Avoiding the use of representation theory in bounding algorithm complexity
5) Using matrix scaling techniques without modification for non-commutative groups
6) Utilizing degenerate spectra to simplify computations
7) Truncating group elements to polynomial bit-length so that runtime remains polynomial
✓ Correct Answer:
The correct answer is 7) Truncating group elements to polynomial bit-length so that runtime remains polynomial.
📚 Reference Text:
at most exponential in the input parameters. This is achieved by relying on Derksen’s degree bounds [Der01] (see Proposition 2.5). 1.7 Additional discussion We would like to point out two important distinctions between the analysis for matrix scaling in Section 1.5 and our analysis here. First is that, as we have seen, there is a major difference betweentheuniformandthenon-uniformversionsofourproblem-whilethiswasnotthecase for matrix scaling. This phenomenon is general and is a distinction between commutative and non-commutative group actions. It has to do with the fact that all irreducible representations ofcommutativegroupsareone-dimensional,whereasfornon-commutativegroupstheyarenot. Secondly, inthematrix scalinganalysis,the upperboundwas easytoobtain aswell. Whereas for us,theupperboundstepisthehardestandrequirestheuseofdeepresultsinrepresentationtheory. The upper bound steps were the cause of main difficulty in the papers [ GGOW16 ,BGO 17,Fra18] aswell /two.superior/three.superiorandthisisonekeypointofdistinctionbetweencommutativeandnon-commutativegroup actions. We believe that our framework of using the highest weight vectors as potential functions for the analysis of analytic algorithms is the right way to approach moment polytope problems - even beyond the cases that we consider in this paper. Theapproachtakenin[ Fra18](forthecaseof d2)isoneofreducingthenon-uniformversion oftheproblemtotheuniformversion,whichwassolvedin[ GGOW16 ]forthecaseof d2(the reduction in [ GGOW17 ] is a simple special case of the reduction in [ Fra18]). The reduction is complicated and a bit ad hoc. We generalize this reduction to our setting ( d¡2) in Section 4, and providingasomewhatmoreprincipledviewofthereductionalongtheway. However,itstillseems rather specialized and mysterious compared to the general reduction in geometric invariant theory from the “non-uniform” to the “uniform” case (also known as the shifting trick , see Section 2.2). Wealso notethat applyingthe resultsof [ BGO 17]to thereduction inSection 4in ablack-box manner does not yield our main theorem (Theorem 1.7) - the number of iterations would be exponential in the bit-complexity ofp, and we would even require an exponential number of bits for /two.superior/three.superiorIn some of the papers, lower bound is the hard step, due to the use of a dual kind of potential function. 14 the randomization step! To remedy these issues with the reduction in Section 4 one must delve intotherelationshipbetweenthereductionandtheinvariantpolynomials. Wewillsee,byfairly involved calculation, that invariant polynomials evaluated on the reduction will result in the same construction of highest weight vectors anyway. This teaches us two lessons: 1.Highest weight vectors are the only suitable potential functions in sight. Though it may have other conceptual benefits, the reduction in Section 4 is no better than the shifting trick for the purpose of obtaining potential functions! 2.WehadtolookattheconstructionofhighestweightvectorsinSection2.4 beforecalculating them from the reduction - the calculation might not have been so easy a priori! Again, the classical construction of highest weight vectors saves the day. ItisinterestingtodiscusssomeofthesalientfeaturesandpossiblevariationsofAlgorithm1 (we expand on these points in the main text): •Iterations and randomness. The algorithm terminates after at most Tpoly maxd i0ni;d;b; 1{" iterations and uses log2pMq polypmaxd i0ni;d;bqbits of randomness. Forfixed oreven inversepolynomial "¡0, this ispolynomial inthe input size. In fact, this is better than the number of iterations in [ Fra18]: there, the number of iterations also depended on ppiq ni 1 . •Bit complexity : Togetanalgorithmwithtrulypolynomialruntime,oneneedstotruncate thegroupelements gpiq’suptopolynomialnumberofbitsafterthedecimalpoint. Weprovide an explanation on why this doesn’t affect the performance of the algorithm in Section 3.3. •Degenerate spectra. Ifpiqis degenerate, i.e. piq jpiq
Question 111multiple-choice
Post-quantum digital signature algorithms are designed to remain secure even when adversaries have access to quantum computers. Advanced approaches increasingly rely on algebraic structures and computational hardness assumptions that differ from classical cryptography.
Which feature directly contributes to both the security and efficiency of a novel post-quantum signature scheme based on non-commutative algebras?
1) Use of integer factorization as the underlying hard problem
2) Reliance on traditional discrete logarithm techniques
3) Implementation on three-dimensional commutative algebras
4) Employment of four-dimensional sparse non-commutative algebras with quadratic equations over finite fields
5) Public key obfuscation solely through hash functions
6) Signature verification using linear equations over real numbers
7) Dependence on classical polynomial-time algorithms for signature generation
✓ Correct Answer:
The correct answer is 4) Employment of four-dimensional sparse non-commutative algebras with quadratic equations over finite fields.
📚 Reference Text:
Title: A NEW CONCEPT FOR DESIGNING POST-QUANTUM DIGITAL SIGNATURE ALGORITHMS ON NON-COMMUTATIVE ALGEBRAS Year: 2022 Paper ID: 955f69b24abc7bf27b89d292b2577fb612068af2 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/955f69b24abc7bf27b89d292b2577fb612068af2 Abstract: Purpose of work is the development of a new approach to designing post-quantum digital signature algorithms that are free from the shortcomings of known analogs – large sizes of the signature and public key. Research method is the use of power vector equations with multiple occurrences of the signature S as a signature verification equation. The computational difficulty of solving equations of the said type relatively the unknown value of S ensures the resistance of the signature scheme to attacks using S as a fitting parameter. The possibility of calculating the value of S by the secret key is provided by using the public key in the form of a set of secret elements of the hidden group, masked by performing left and right multiplications by matched invertible vectors. Results of the study include a new proposed concept for the development of post-quantum digital signature algorithms on non-commutative algebras, which use a hidden commutative group. One of its main differences is the use of a secret key in the form of a set of vectors, the knowledge of which makes it possible to calculate the correct signature value for the random powers present in the verification equation. The form of the latter defines a system of quadratic vector equations connecting the public key with the secret, which is reduced to a system of many quadratic equations with many unknowns, given over a finite field. The computational difficulty of finding a solution to the latter system determines the security of the algorithms developed within the framework of the proposed concept. A quantum computer is ineffective for solving this problem, therefore, the said algorithms are post-quantum. As analogs in construction, digital signature algorithms based on the computational difficulty of the hidden discrete logarithm problem are considered, however, the use of a hidden group and exponentiation operations represent only a general technique for ensuring the correctness of the signature schemes developed within the framework of the concept, and not for specifying a basic computationally difficult problem. To improve the performance of the signature generation and verifications procedures, the four-dimensional algebras defined by sparse basis vector multiplication tables are used as an algebraic support. The proposed concept is confirmed by the development of a specific post-quantum algorithm that provides a significant reduction in the size of the public key and signature in comparison with the finalists of the NIST global competition in the nomination of post-quantum digital signature algorithms. Practical relevance: The developed new concept for constructing post-quantum digital signature algorithms expands the areas of their application in conditions of limited computing resources
Question 112multiple-choice
The hidden subgroup problem (HSP) is a central challenge in quantum computing, underpinning efficient solutions to key computational problems. Quantum measurement strategies and group structure play crucial roles in designing algorithms for both abelian and nonabelian groups.
Which of the following statements accurately describes a significant advance in solving the hidden subgroup problem for nonabelian groups using quantum algorithms?
1) Efficient quantum algorithms for the dihedral group HSP have been discovered, enabling polynomial-time solutions to lattice problems.
2) The pretty good measurement (PGM) is always optimal for all nonabelian groups when distinguishing hidden subgroups.
3) Entangled measurements are unnecessary for efficiently solving nonabelian hidden subgroup problems with quantum algorithms.
4) Efficient quantum algorithms have been developed for nonabelian groups such as metacyclic groups and groups of the form Z_r^p ⋊ Z_p, leveraging optimal measurements and entanglement.
5) The HSP for symmetric groups has been solved in polynomial time, leading to efficient algorithms for graph isomorphism.
6) Shor’s algorithm provides efficient solutions to both abelian and nonabelian hidden subgroup problems.
7) Subexponential time algorithms for the dihedral group HSP rely exclusively on classical computation.
✓ Correct Answer:
The correct answer is 4) Efficient quantum algorithms have been developed for nonabelian groups such as metacyclic groups and groups of the form Z_r^p ⋊ Z_p, leveraging optimal measurements and entanglement..
📚 Reference Text:
arXiv:quant-ph/0504083v2 26 Apr 2005From optimal measurement to efficient quantum algorithms for the hidden subgroup problem over semidirect product groups Dave Bacon∗ dabacon@santafe.eduAndrew M. Childs† amchilds@caltech.eduWim van Dam‡ vandam@cs.ucsb.edu Abstract We approach the hidden subgroup problem by performing the so-ca lled pretty good mea- surement on hidden subgroup states. For various groups that ca n be expressed as the semidirect product of an abelian group and a cyclic group, we show that the pre tty good measurement is optimal and that its probability of success and unitary implementatio n are closely related to an average-case algebraic problem. By solving this problem, we find effic ient quantum algorithms for a number of nonabelian hidden subgroup problems, including some for which no efficient algorithm was previously known: certain metacyclic groups as well as all groups of the form Zr p⋊ Zpfor fixedr(including the Heisenberg group, r= 2). In particular, our results show that entangled measurements across multiple copies of hidden subg roup states can be useful for efficiently solving the nonabelian hsp. 1 Introduction The hidden subgroup problem ( hsp) stands as one of the major challenges for quantum compu- tation. Shor’s discovery of an efficient quantum algorithm fo r factoring and calculating discrete logarithms [30], which essentially solves the abelian hsp[4, 19], focused attention on the question of what computational problems might be solved asymptoticall y faster by quantum computers than by classical ones. In particular, we would like to understan d when the nonabelian hidden subgroup problem admits an efficient quantum algorithm. Considerable progress on this question has been made since S hor’s discovery. Efficient quantum algorithms have been found for the case where the hidden subg roup is promised to be normal and there is an efficient quantum Fourier transform over the group [13], or where the group is “almost abelian” [12] or, more generally, “near-Hamiltonian” [11] . In addition, efficient algorithms have been found for several groups that can be written as semidire ct products of abelian groups: the wreath product Zn 2≀Z2[28], certain groups of the form Zn pk⋊Z2for a fixed prime power pk[10], q-hedral groups with qsufficiently large [21], and particular groups of the form Zpk⋊Zpwithpan odd prime [17]. Unfortunately, efficient algorithms have been elusive for tw o cases with particularly significant applications: the dihedral group and the symmetric group. A n efficient algorithm for the hspover the symmetric group would lead to an efficient algorithm for gr aph isomorphism [4, 7, 3, 16], and an efficient algorithm for the dihedral hsp(based on the standard approach described in Section 2.1) would lead to an efficient algorithm for certain lattice probl ems [26]. While no polynomial-time quantum algorithms for these problems are known, Kuperberg recently gave a subexponential (but superpolynomial) time and space algorithm for the dihedral hsp[20], and Regev improved the space requirement to be only polynomial [27]. ∗Santa Fe Institute, Santa Fe, NM 87501, USA †Institute for Quantum Information, California Institute o f Technology, Pasadena, CA 91125, USA ‡Department of Computer Science, University of California, Santa
Question 113multiple-choice
Quantum phases on graphs expand the landscape of topological order by allowing for unconventional connectivity and topology, particularly when modeled with abelian gauge theories. Ground state degeneracy and entanglement entropy play key roles in distinguishing these phases and their potential applications in quantum information science.
Which property best distinguishes a topologically ordered phase on a graph from one lacking topological order when using abelian gauge theory models?
1) Absence of long-range entanglement in all ground states
2) Ground state degeneracy that varies only with system size
3) Local order parameters sufficient to classify all phases
4) Presence of gapless excitations throughout the spectrum
5) A constant contribution to entanglement entropy reflecting non-local quantum correlations
6) Spontaneous breaking of global symmetries
7) Ground state uniqueness irrespective of topology
✓ Correct Answer:
The correct answer is 5) A constant contribution to entanglement entropy reflecting non-local quantum correlations.
📚 Reference Text:
Title: Novel quantum phases on graphs using abelian gauge theory Year: 2021 Paper ID: 9b3c72d3dc7218ee388f5f4aa76b8b7e167201e1 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/9b3c72d3dc7218ee388f5f4aa76b8b7e167201e1 Abstract: Graphs are topological spaces that include broader objects than discretized manifolds, making them interesting playgrounds for the study of quantum phases not realized by symmetry breaking. In particular they are known to support anyons of an even richer variety than the two-dimensional space. We explore this possibility by building a class of frustration-free and gapped Hamiltonians based on discrete abelian gauge groups. The resulting models have a ground state degeneracy that can be either a topological invariant, an extensive quantity or a mixture of the two. For two basis of the degenerate ground states which are complementary in quantum theory, the entanglement entropy (EE) is exactly computed. The result for one basis has a constant global term, known as the topological EE, implying long-range entanglement. On the other hand, the topological EE vanishes in the result for the other basis. Comparisons are made with similar occurrences in the toric code. We analyze excitations and identify anyon-like excitations that account for the topological EE. An analogy between the ground states of this system and the θ-vacuum for a U(1) gauge theory on a circle is also drawn.
Question 114multiple-choice
Quantum Fourier Transform (QFT) is a fundamental operation in quantum computing, enabling analysis and extraction of frequency components from quantum-encoded signals such as audio data. Simulating QFT circuits on classical hardware allows researchers to visualize frequency information and test quantum algorithms for signal processing applications.
When a quantum circuit implementing the Quantum Fourier Transform is applied to qubits encoding sampled audio amplitudes and followed by measurement in the computational basis, what does the amplitude of each basis state after measurement most directly represent?
1) The phase shift introduced by each quantum gate in the circuit
2) The probability of observing a specific time-domain sample
3) The total energy of the quantum system
4) The strength of each frequency present in the original signal
5) The number of qubits used in the encoding
6) The fidelity of the quantum simulation
7) The error rate from gate imperfections
✓ Correct Answer:
The correct answer is 4) The strength of each frequency present in the original signal.
📚 Reference Text:
QFT to quantum circuit, create measurement register, and draw visualization of circuit 2qft(audio qc, n qubits) 3audio qc.measure all() 4 5#Simulation 6sim = Aer.get backend(”aer simulator”) 7qobj = assemble(audio qc) 8counts = sim.run(qobj).result().get counts() 9qftcounts = get qftcounts(counts, n qubits)[: len(samples)//2] 10plothistogram(counts) Figure 22 Applying the QFT Since each qubit is measured in the Pauli-Z basis, all qubits will collapse into one of the two eigenstates of the Pauli-Z operator probabilities dictated by the superposition that we put the qubits in. The QFT on each qubit in a superposition has the e ect of causing the qubits collapse in a way that gives us the Fourier transform of the input amplitudes. 23 SciPost Physics Codebases Submission Figure 23 Histogram of the QFT Output of A440 5.1.5 Decoding the Result If the nal output of the state is given by : |ψ/angbracketright=/summationdisplay iai|i/angbracketright (52) whereiare the basis states, then the information we need to determine the frequency of the musical note is stored in the amplitude in ai. Here we will use Python to evaluate the top indices of our output. To get the value of frequency we have to multiply these peaks by framerate 2nqubitsHz. Python: 1#Calculate Frequencies and print top 2 frequencies 2topindices = np.argsort( −np.array(qft counts)) 3freqs = top indices ∗frame rate/2 ∗∗nqubits 4print (freqs[:2]) Figure 24 Obtaining the Frequency The top two indices computed for frequency were computed to be: [430.6640625,473.73046875] (53) 24 SciPost Physics Codebases Submission These were the top two frequencies output by our QFT circuit. These are very close to the actual frequency which we know to be 440Hz since we sampled the musical note A440. 5.2 Using the QFT on a Musical Chord Now that we have demonstrated using the QFT to nd a single musical note, the next step is to determine if we can use this method to nd multiple musical notes played together. Multiple notes played at the same time is known as a musical chord. Below is a F-major chord comprised of three notes: C3, F3, and A4 all played simultaneously. The resulting audio signal will be three frequencies in superposition. C3 corresponds to 130.81Hz, F3 to 174.61Hz, and A4 to 440.0Hz. Figure 25 F-Major Chord Following the same steps on the single note example, we can sample the audio of this chord played and obtain an array of amplitudes for each sample and plot the results. Figure 26 Output Audio Waveform of F-Major Chord One might notice that it would be dicult to discern the three distinct frequencies of the musical chord by looking at the graph alone to calculate the period or frequency. The QFT is useful here because it will transform this data from the time domain to the frequency domain where we will see amplitudes of the three individual frequencies clearly. Next, we encode the samples again onto our quantum system and apply the QFT. After simulating the circuit and outcome we are left with the following results. 25 SciPost Physics Codebases Submission Figure 27
Question 115multiple-choice
Quantum computing enables the development of new algorithms for efficient data processing, including image and signal manipulation, by leveraging quantum parallelism and specialized transformations. Understanding the characteristics and limitations of these quantum algorithms is crucial for advancing practical applications.
Which statement best describes a key advantage of quantum image interpolation algorithms over classical counterparts?
1) Quantum image interpolation algorithms rely exclusively on entanglement for efficiency.
2) They require exponentially more resources as image size increases.
3) Their computational complexity remains constant regardless of image size due to processing image subspaces in quantum superposition.
4) They cannot utilize techniques inspired by classical image processing such as compression.
5) They are limited to processing only classical data encoded in quantum circuits.
6) Quantum parallelism does not contribute to any speedup in image interpolation tasks.
7) These algorithms can only operate on fully decohered quantum states.
✓ Correct Answer:
The correct answer is 3) Their computational complexity remains constant regardless of image size due to processing image subspaces in quantum superposition..
📚 Reference Text:
be efficiently resam- pledviainterpolationalgorithms. Itcanalsobeextended to any uploading technique that deals with non-classical band-limited quantum signals. Additionally, this inter- polation technique can be used on genuinely quantum states, but the interpolation will be extended along com- putational basis states which might fail to capture corre- lations that go beyond that. We have also showcased the power of processing quan- tum data in superposition. Encoding different sets ofdata using label ancillas allows for the implementation of quantum transformations in parallel. Furthermore, by extending native ideas of natural image processing, used in the JPEG procedure for image compression, we can employ this parallel processing power of a quantum circuit to gain a substantial advantage. By processing subspaces of an image simultaneously using quantum su- perposition, wecanperforminterpolationinacomplexity that is constant with the original size of the image. We would like to further highlight the implementation of efficient quantum transformations in order to provide quantum advantages to algorithms that might not ini- tially rely on them. After all, at the core of the efficiency of Shor’s factorization algorithm [34] is the use of the QFT. Looking into areas where these types of transfor- mations are extensively used may result in further av- enues for quantum advantage. Natural image and signal processingarefieldsthathaveevolvedaroundsuchtrans- formations, furtherworkmightborrowfromsuchwellun- derstood fields in order to enhance quantum algorithms in different ways or explore interpolation techniques with even more advanced transformations. Compression tech- niques that rely on these transformations might also pro- vide quantum advantages. The code used to simulate the quantum circuits pre- sented is available online [35]. ACKNOWLEDGEMENTS The author would like to acknowledge J. I. Latorre and I. Roth for fruitful discussions. 7 [1] J. J. García-Ripoll, Quantum-inspired algorithms for multivariate analysis: from interpolation to partial dif- ferential equations, Quantum 5, 431 (2021). [2] P. García-Molina, J. Rodríguez-Mediavilla, and J. J. García-Ripoll, Quantum fourier analysis for multivariate functions and applications to a class of schrödinger-type partial differential equations, Physical Review A 105, 012433 (2022). [3] N. Ahmed, T. Natarajan, and K. R. Rao, Discrete co- sine transform, IEEE transactions on Computers 100, 90 (1974). [4] G. K. Wallace, The jpeg still picture compression stan- dard,IEEEtransactionsonconsumerelectronics 38,xviii (1992). [5] S. Efthymiou, S. Ramos-Calderer, C. Bravo-Prieto, A. Pérez-Salinas, D. García-Martín, A. Garcia-Saez, J. I. Latorre, and S. Carrazza, Qibo: a framework for quan- tum simulation with hardware acceleration, Quantum Science and Technology 7, 015018 (2021). [6] S. Efthymiou, S. Carrazza, C. Bravo-Prieto, Adrian- PerezSalinas, S. Ramos, D. García-Martín, M. Lazzarin, N. Zattarin, A. Pasquale, Paul, and J. Serrano, qi- boteam/qibo: Qibo 0.1.7 (2022). [7] L. Grover and T. Rudolph, Creating superpositions that correspond to efficiently integrable probability distribu- tions, arXiv preprint quant-ph/0208112 (2002). [8] V. V. Shende, S. S. Bullock, and I. L. Markov, Syn- thesis of quantum-logic circuits, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Sys- tems25, 1000 (2006). [9] A. Kitaev and W. A. Webb, Wavefunction prepara- tion and resampling using a quantum computer, arXiv preprint arXiv:0801.0342 (2008). [10] M. Plesch and Č. Brukner, Quantum-state preparation with
Question 116multiple-choice
The group SU(2) plays a crucial role in quantum mechanics, especially for describing the symmetries of spin systems and the transformations of qubit states. Its representation theory connects mathematical structure to physical phenomena such as spin and quantum state rotations.
Which property of SU(2) is directly responsible for the fact that a spin-½ particle requires a 720° rotation to return to its original quantum state, rather than 360°?
1) SU(2) is composed only of diagonal matrices
2) The Pauli matrices commute under multiplication
3) SU(2) is isomorphic to SO(3)
4) The trivial representation of SU(2) acts on all group elements
5) SU(2) is a double cover of SO(3), leading to the spinor sign flip after a 360° rotation
6) SU(2) matrices have purely imaginary eigenvalues
7) The fundamental representation of SU(2) acts on three-dimensional vectors
✓ Correct Answer:
The correct answer is 5) SU(2) is a double cover of SO(3), leading to the spinor sign flip after a 360° rotation.
📚 Reference Text:
unitary group SU(2), and the QML task of classifying single-qubit states according to their purity (see Fig. 2(b)). Our choice of considering SU(2)is further motivated both for its privileged posi- tion in quantum physics as the symmetry group underlying spins and for its remarkably well behaved representation theory. Again, just like with discrete groups, we always have the trivial representation of Gon any vector space V, given byRg=12GL(V)for allg2G. All matrix Lie groups also have a so-called fundamen-Example 3: SU(2)and its fundamental rep (spin-1/2) Recall that the special unitary group is defined as SU(2) =fU2GL(C2) :UyU=1and det(U) = 1g: Take 1 qubit V=C2. The fundamental represen- tationU:G!GL(V)is given by Ug=g, i.e., the matrix in Gis the same as its representative. While we are here, it is good to recall that any ma- trixUg2SU(2)can be expressed in terms of Pauli matricesX;Y;Zby Ug=c01+i(c1X+c2Y+c3Z); where (c0;c1;c2;c3)is a real vector with Euclidean normj(c0;c1;c2;c3)j= 1. Note that since this de- fines an invertible and continuous map between the 3-sphereS3R4andSU(2), the two spaces are homeomorphic. As special cases of this decomposi- tion, we have the rotations around Cartesian axes of angleof the Bloch sphere: RX() =e iX= 2= cos(=2)1 isin(=2)X RY() =e iY=2= cos(=2)1 isin(=2)Y RZ() =e iZ=2= cos(=2)1 isin(=2)Z tal representation ordefining representation6(Example 3), which means that the matrices in the group and the ma- trices in the representation coincide (up to isomorphism, which we will learn morally means “up to change of basis” in Definition 16). Try to note this important distinction: even though the matrices between the group Gand its rep- resentativesfRg:g2GgGL(V)are identical, we think of the abstract group and its representatives as conceptu- ally distinct. For instance, spin representations of angular momentum are, at their core, (irreducible) representations of the Lie group SU(2). In the case of spin- 1=2, this is ex- 6There yet another unfortunate clash of terminology here–many physicists commonly use these terms interchangeably, but mathe- maticians would say that the definition here is strictly the defining representation, which for classical matrix Lie groups is one of the fundamental representations. Matrix Lie groups commonly have more fundamental representations than the defining rep (e.g. the defining rep 3and its dual 3forSU(3)), but in the case of SU(2), the defining representation is the only fundamental representation. 12 actly the fundamental representation of SU(2)onV=C2 whereboththematricesandtheirrepresentativesareofthe forme i ~where= (x;y;z)is a vector of real angles and= (X;Y;Z )is a vector whose entries are the Pauli matrices. But in higher spin, e.g., spin- 1, the group SU(2) is fixed but the representation V=C3is not, and so the 33representative “spin matrices” take on the new form SX;SY;SZ. More light will be shed on the spin representa- tions and angular momentum once we get to Lie algebras: a pervading theme of Lie theory is that it is often easier to work on the Lie algebra (e.g., the spin matrices X;Y;Z with commutation relations) than on the Lie group (here, the rotation matrices e i). Herewenotethatthefactthattheparameter isdivided by a factor of two in Example 3 is closely related to an im- portant topological fact: SU(2)is the
Question 117multiple-choice
Quantum algorithms leveraging group symmetries have become central to solving complex problems in quantum many-body physics, especially when classical computational resources are insufficient. The J1-J2 Heisenberg model is a key example where the interplay of frustration and symmetry determines both computational complexity and the nature of quantum phases.
Which statement accurately describes why quantum speedup is expected to persist for solving highly frustrated regimes of the J1-J2 Heisenberg model associated with symmetric group operations?
1) Because classical FFT algorithms for Sn scale polynomially with the number of spins, making simulation straightforward.
2) Because quantum circuits for Sn always scale exponentially regardless of model complexity.
3) Because classical dequantization techniques can efficiently simulate all quantum time evolutions in group-theoretic models.
4) Because all ground states of frustrated J1-J2 models can be analytically constructed via Marshall-Lieb-Mattis theorems.
5) Because the regular representation of Sn admits low-dimensional classical embeddings for highly frustrated quantum phases.
6) Because classical algorithms exploiting Schur-Weyl duality achieve superexponential speedup over quantum algorithms in the frustrated regime.
7) Because classical simulation of quantum time evolution involving many Sn group elements in the highly frustrated regime remains intractable, ensuring quantum speedup persists.
✓ Correct Answer:
The correct answer is 7) Because classical simulation of quantum time evolution involving many Sn group elements in the highly frustrated regime remains intractable, ensuring quantum speedup persists..
📚 Reference Text:
classically difficult question and the best classical algorithms Sn-FFT requires a factorial complexity [ 35,36]a sSnhasn! group elements and so is the dimension of its regular represen- tation (see the Wedderburn theorem in the Appendix for more details). Therefore, comparing with the complexity ofSn-FFT, there is a superexponential quantum speed per query. However as a caveat, the entire Hilbert space of nqudits only scales exponentially with nand Snirreps decomposed from the system by Schur-Weyl duality also scales exponentially. Therefore, it would be more reason- able and cautious to refer to a superpolynomial quantum speedup for Sn-CQA. Except comparing with Sn-FFT, recent work from Refs. [ 10,81] proposes the notion of dequantization to compare the efficiencies of classical and quantum algo- rithms. Roughly speaking, with well-prepared quantum initial states, quantum algorithms can always be exponen- tially faster than the best counterpart classical algorithms. Assume classical algorithms also have efficient access to input. If the output can now be evaluated with at most polynomially larger query complexity then the quantum analogy, it is said to be dequantized with no genuine quantum speedup. In our case, let us assume our initial states—Schur-basis elements |αT,μS/angbracketrightor their linear com- binations can be efficiently accessed with classical meth- ods. Even though, dequantization still unlikely happens. Except conducting Sn-FFT, matrix representations of σ∈ Sncan also be efficiently sampled [ 48,75], but the method works exclusively for a single group element. To sample U(p) CQA(θ)|αT,μS/angbracketrightprocessed after Sn-CQA from Eq. (3), the time evolution of CQA Hamiltonians is expanded and approximated by at least superpolynomially many Sngroup elements (see the Appendix for more details) and hence is still thought to be classically intractable. V.C[Sn] SYMMETRIES OF THE J1-J2 HEISENBERG HAMILTONIAN The spin-1 /2J1-J2Heisenberg model is defined by the Hamiltonian: ˆHp=J1/summationdisplay /angbracketleftij/angbracketrightˆSi·ˆSj+J2/summationdisplay /angbracketleft/angbracketleftij/angbracketright/angbracketrightˆSi·ˆSj,( 8 ) where ˆSi=(ˆSx i,ˆSy i,ˆSz i)represents the spin operators at site iof the concerned lattice. The symbols /angbracketleft ···/angbracketright and /angbracketleft/angbracketleft··· /angbracketright/angbracketright indicate pairs of nearest- and next-nearest-neighbor sites, respectively. The J1-J2model has been the subject of intense research over its speculated novel spin-liquidphases at frustrated region [ 82]. The unfrustrated regime (J 2=0o r J1=0)for the antiferromagnetic Heisenberg model is characterized by the bipartite lattices, for which the sign structures of the respective ground states areanalytically given by the Marshall-Lieb-Mattis theorem [54–56]. As an important result, ground states of unfrus- trated bipartite models are proven to live in the Snirrep corresponding to the Young diagram λ=(n/2,n/2).B y Schur-Weyl duality, this subspace is often referred to as the direct sum of SU (2)-invariant subspaces with total spin J=0 in the context of physics [cf. Figs. 2(a) and2(b)]. With this fact, algorithms like Ref. [ 38] has been designed to enforce SU (2)symmetry at J=0 and solve Heisenberg models without frustration. The system is known to be highly frustrated when J1andJ2are comparable J2/J1≈0.5 [ 83] and near the region of two phase transitions from Neel ordering to the quantum paramagnetic phase and from quantum para- magnetic to colinear phase, where no exact solution is known. Moreover, little
Question 118multiple-choice
In quantum information theory, locally maximally entangled (LME) states are pure quantum states in multipartite systems where every subsystem appears maximally mixed when others are traced out. The existence and structure of such states depend critically on the dimensions and number of subsystems involved.
In which scenario does the dimension of the space of locally maximally entangled states modulo local unitary transformations and permutations exceed 3, indicating a rich variety of entanglement structures?
1) A system of five qubits
2) A system of three qubits
3) A system of two qutrits
4) A system of three subsystems each of dimension 2
5) A bipartite system of subsystems with unequal dimensions
6) A system of three subsystems each of dimension 3
7) A system of four qubits
✓ Correct Answer:
The correct answer is 1) A system of five qubits.
📚 Reference Text:
dimension of the space SLME/K∼P(H)//G is larger than 3. 9 For qubit systems, this holds whenever n>4, so we reproduce the results in [GKW17]. For systems ofnqutrits, this holds for n>3. The condition also holds for any system of three or more subsystems of the same dimension d>3. These cases reproduce and generalize the results in [SWGK17]. Our general condition covers all remaining cases , where the subsystem dimensions are not equal. 2 Setup and simple cases We will consider pure states in a multipart Hilbert space H=H1⊗···⊗H n with subsystems Hiof dimension di. We can write a general state explicitly as |Ψ/an}b∇acket∇i}ht=/summationdisplay iψi1···in|i1/an}b∇acket∇i}ht⊗···⊗|in/an}b∇acket∇i}ht, (2.1) where states |ik/an}b∇acket∇i}htwith 1≤ik≤dkform an orthonormal basis of Hk. The density matrix for thekth subsystem is given by ρk= tr¯k|Ψ/an}b∇acket∇i}ht/an}b∇acketle{tΨ|or explicitly as ρjklk=/summationdisplay iψi1···jk···in(ψ∗)i1···lk···in(2.2) A locally maximally entangled state |Ψ/an}b∇acket∇i}htis defined as a state such that for every kwe have ρjklk=1 dkδjklk. (2.3) 2.1 The Schmidt decomposition, bipartite systems, general nec- essary conditions For a bipartite system with H=HA⊗ H¯A, we can write a general (pure) state using the Schmidt decomposition as |Ψ/an}b∇acket∇i}ht=/summationdisplay i√pi|ψA i/an}b∇acket∇i}ht⊗|ψ¯A i/an}b∇acket∇i}ht, (2.4) where|ψA i/an}b∇acket∇i}htare orthonormal states in HA,|ψ¯A i/an}b∇acket∇i}htare orthonormal states in H¯A, andpiare positive real numbers with/summationtext ipi= 1. The density operator for the first subsystem is ρ1=/summationdisplay ipi|ψA i/an}b∇acket∇i}ht/an}b∇acketle{tψA i|. (2.5) This is a multiple of the identity operator if and only if {|ψA i/an}b∇acket∇i}ht}form an orthonormal basis of HAandpi=1 dAfor alli. This is possible only if the dimension of HAis less than or equal 10 to the dimension of H¯A. Otherwise, we can’t have the required dAorthogonal states {|ψA i/an}b∇acket∇i}ht} since the number of these is equal to the number of orthonormal s tates|ψ¯A i/an}b∇acket∇i}htwhich is limited by the dimension of H¯A. We can consider a general system with dimensions ( d1,...,dn) as a bipartite system with HA=HkandH¯A=⊗i/ne}ationslash=kHi. Then a state |Ψ/an}b∇acket∇i}ht ∈ H=⊗Hican be LME only if ρk≡ρAis proportional to the identity; we have just seen that this requires dA≤d¯A, so we arrive at general necessary conditions dk≤/productdisplay i/ne}ationslash=kdi (2.6) for the existence of locally maximally mixed states. It has been sugge sted that these condi- tions are also sufficient, but we will see that this is not the case. 2.2 Explicit construction: n= 2 For a two-part system, the necessary conditions (2.6) give immedia tely thatd1=d2≡din order for there to exist LME states. The discussion in the previous subsection also implies that such a state can be written via the Schmidt decomposition as |Ψ/an}b∇acket∇i}ht=1√ d/summationdisplay i|ψ1 i/an}b∇acket∇i}ht⊗|ψ2 i/an}b∇acket∇i}ht (2.7) where|ψ1 i/an}b∇acket∇i}htand|ψ2 i/an}b∇acket∇i}htare orthonormal bases of H1andH2. Any state of this form is in SLME. The group of local unitary transformations in K= SU(d)×SU(d) allow us to independently rotate the bases for the two subsystems, so any LME state (2.7) can always be written as U1⊗U2|Ψ/an}b∇acket∇i}htBellwhere|Ψ/an}b∇acket∇i}htBellis the Bell state (1.2). We thus have the well-known result that forn= 2, LME states exist if and only if d1=d2, and in this case, the Bell state is the unique LME state up to local unitary transformations. 2.3 Explicit
Question 119multiple-choice
In finite group theory, the study of character tables is crucial for understanding the representations and structure of complex groups, including non-split extensions. The group 26.S8, arising as a maximal subgroup of certain orthogonal groups over finite fields, is an example of such an intricate construction.
Which subgroup serves as one of the inertia factors for the irreducible characters of the non-split group extension 26.S8?
1) S7 × S1
2) S5 × S3
3) S4 × S2
4) (S4 × S4):2
5) S3 × S6
6) S2 × S8
7) S6 × S4
✓ Correct Answer:
The correct answer is 4) (S4 × S4):2.
📚 Reference Text:
Title: On the character table of non-split extension 26.S8 Year: 2019 Paper ID: 6abeaf65ac4f6ee43cc2c96a8eb682035fd4efb1 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/6abeaf65ac4f6ee43cc2c96a8eb682035fd4efb1 Abstract: Problem Statement & Objective: Character tables of maximal subgroups of finite simple groups provide considerable amount of information about the groups. In the present article, our objective is to compute the character table of one maximal subgroup of the orthogonal group PSO8+(3). Approach: The projective special orthogonal group PSO8+(3)≅O8+(3).21 is obtained from the special orthogonal group SO8+(3) on factoring by the group of scalar matrices it contains. The group O8+(3).21 has a maximal subgroup of the form 26.S8 with index 3838185. The group Q ≅ 26 · S8 is a non-split group extension of an elementary abelian 2-group of order 64 by the symmetric group S8. We apply the Fischer-Clifford theory to compute the irreducible characters of the extension 26 · S8. Results and Conclusion: We produce 64 conjugacy classes of elements as well as 64 irreducible character of the non-split group extension 26 · S8 corresponding to the three inertia factors H1 = S8, H2 = S6 × 2 and H3 = (S4 × S4):2.
Question 120multiple-choice
Quantum annealing is a computational approach used to solve optimization problems by leveraging quantum mechanical effects, often compared to classical methods like simulated annealing. The implementation of time evolution in quantum simulations requires specialized techniques depending on hardware architecture.
Which method allows quantum annealers to simulate real-time dynamics by encoding time evolution as an optimization problem over an enlarged Hilbert space?
1) Trotterization of the time evolution operator
2) Euclidean lattice discretization
3) Classical Monte Carlo sampling
4) Imaginary-time propagation
5) Feynman clock states with ancillary encoding
6) Grover's algorithm for state preparation
7) Adiabatic quantum gate decomposition
✓ Correct Answer:
The correct answer is 5) Feynman clock states with ancillary encoding.
📚 Reference Text:
theseintoourvariationalcalculation,therhsofEq.( 15)canrewrittenasacostfunction F=/angbracketleftψ|ˆH|ψ/angbracketright–η/angbracketleftψ|ψ/angbracketright=Nconf/summationdisplay α,βK/summationdisplay i,jQαβ,ijqα,iqβ,j, (17) Frommetal. EPJQuantumTechnology (2023) 10:31 Page8of19 where Qαβ,ij=2i+j–2K–2z(–1)δiK+δjKhαβ+δαβδij˜Qα,i, ˜Qα,i=2i–K–z+1(–1)δiKNconf/summationdisplay γa(z) γhγα, (18) hαβ=Hαβ–ηδαβ. Here, ηrepresents the tunable parameter multiplying the penalty term. In addition to givingthevariationalparametersafloating-pointrepresentation,intheabovedefinitionswe have already introduced the parameter z, which is used in the adaptive variational search method [ 10]. This procedure iteratively improves the estimates for the a (z+1) αby distributingthe KsamplingpointsaroundtheprecedingsolutiontotheQUBOproblem, a(z) α, a(z+1) α=a(z) α–qα,K 2z+K–1/summationdisplay i=1qα,i 2K–i+z, (19) startingat a(0) α=0.Theestimatefortheeigenstateisrefinedateachstep,hencethename “zooming” for this procedure. The number of zoom steps plays a significant role in the overall computational cost of our calculation, as each refinement requires calls to the quantumannealer. The quantum computations are done on the quantum annealing hardware Advan- tage_system5.1 fromD-Wave[ 11],whichisaccessibleviaitsAPID-WaveOcean[ 21]. Asoursystemsizesarestillsmall(c.f.Table 1),theresultscanbecomparedwiththeexact solutionusingtheHamiltonianEq.( 3)aswellaswithsimulatedannealingviatheOcean packageneal. As alternative, which we did not employ, D-Wave offers hybrid solvers s.a.theKerberosSampler [21]thatattempttobreakdowntheoriginalQUBOmatrix into smaller pieces to be subsequently solved using classical or quantum hardware. This appears to be particularly useful for system sizes that cannot be embedded on currentlyavailableannealerarchitectures. It should be noted that for computations employing both simulated and quantum an- nealing,resultsstillhavea ηdependence,seeEq.( 18).Asalreadynoticedin[ 9,10],conver- gencetothetrueground-stateisachievedfor ηlyingintheactualvicinityoftheground- state energy, E 0(approaching from above). For practical reasons, one can determine the “suitable” ηfor a given Qby solving Eq. ( 18)i t e r a t i v e l yi n η, terminating the calculation when a certain convergence criterion, s.a. relative improvement in the solution, is ful- filled.Thisstrategyworkswellforlocalcomputationswithsimulatedannealingandcouldinprinciplebeemployedalsoforquantumannealing.Here,runtimeonthequantuman-nealeristhemajorconstraint. We finally comment on our setup when accessing the annealer via the provided soft- ware package. We use the quantum annealer in its forward annealing mode with defaultannealing schedule and annealing time t f=20μs. At least one more parameter needs to be provided by the user during the quantum annealing computations. This is what is re- ferred to as the chain strength. For our calculations, we find automatic chain strengthtuning (default option) to be sufficient. Figure 3shows results from both simulated and Frommetal. EPJQuantumTechnology (2023) 10:31 Page9of19 Figure3 (Left):Ground-stateexpectationvaluesfor D3forthefullKogut-SusskindHamiltonian(red),electric term(black),andmagneticterm(blue)asafunctionoftheinverseHamiltoniancouplingsquared.Theopen symbolsrepresenttheminimumresultfromthequantumannealerwhilethefilledsymbolswereobtained fromclassicalsimulatedannealing.Thecoloredband,althoughbarelyvisible,representsthemeanand samplestandarddeviationofthemeasurementsfromthequantumannealer.Simulationparametersforthe latterwere K=3withzmax=5zoomstepsand nreads=1000.(Right):Thesamequantitiesfor G=D4where thesamplestandarddeviationfromQAismuchlarger.Thisisduetothefactthatmorecomputingresources areneededtoaccuratelydeterminetheminimum.SimulationparametersforQAwere K=2withzmax=7 zoomstepsand nreads=2000 quantumannealing,fortheground-stateenergy /angbracketleftH/angbracketright(red)aswellastheexpectationvalue for the magnet part /angbracketleftHB/angbracketright(blue)and kinetic part /angbracketleftHE/angbracketright(black)for G=D3(left).When go- ing toG=D4(right), an increase in computational resources is needed due to the more complicatedenergylandscapeforthelargergroup,whichwehoweveronlypartiallymeetduetoruntimerestrictions. 3.2 Timeevolution OneofthemainmotivationsforworkingintheHamiltonianformulationoflatticegaugetheoriesistheabilitytoaccessreal-timedynamics.Thisstandsinstarkcontrasttomain-stream lattice calculations which work in Euclidean space and must perform an analyticcontinuationofnumericaldatainordertoaccessreal-timephysics.Inthegate-basedap-proach, the so-called Trotter approximation is employed to the time evolution operator, ˆU(T)=exp{–iTˆH},whichevolvesaninitialstatebyafinitetime T[15].Thisapproxima- tionconsistsofreplacingthefull ˆUwithproductsofoperatorswhichevolvethesystemon a smaller time interval, δt. Corrections to this approximation typically scale with powers ofδt.Thisapproachtotime-evolutionofquantumstatesallowsforanefficientsimulation ofthetheoryusinguniversalquantumcomputers. In order to solve this problem on the quantum annealer, however, one must reformu- late time evolution as an optimization problem. This can be done by the introduction ofFeynman clock states [ 22], a mechanism first applied to quantum chemistry calculations in order to generate parallel-in-time quantum dynamics [ 23]. We thus have to introduce anancillaryquantumsystemwithstates |t/angbracketright,t=1,2,..., N twhereNtisthenumberoftime- slices in the time evolution. Tensoring this orthonormal state with our as-yet-unknownstatevector |ψ t/angbracketrightateachtimeslice,theproblemoftimeevolutionisequivalenttothemin- imizationofthefollowingfunctional L=Nt/summationdisplay t,t/prime=1/angbracketleftt/prime|/angbracketleftψt/prime|ˆC|ψt/angbracketright|t/angbracketright–η/parenleftBiggNt/summationdisplay t,t/prime=1/angbracketleftt/prime|/angbracketleftψt/prime|ψt/angbracketright|t/angbracketright–1/parenrightBigg , (20) Frommetal. EPJQuantumTechnology (2023) 10:31 Page10of19 where ˆC≡ˆC0+1 2/parenleftbig I⊗|t/angbracketright/angbracketleftt|+I⊗|t+1/angbracketright/angbracketleftt+1| –ˆUδt⊗|t+1/angbracketright/angbracketleftt|–ˆU† δt⊗|t/angbracketright/angbracketleftt+1|/parenrightbig . (21) Hereδt≡T/Ntisthestepsizeintime, ηisaLagrangemultiplieranalogoustoourprevious
Question 121multiple-choice
Quantum algorithms for estimating Hamiltonian eigenvalues rely on efficient resource management, including the elimination of unwanted ancillary qubits (“garbage”) and complex phase factors. Advanced techniques use mathematical lemmas to refine both the output quality and the query complexity of these estimators.
In constructing a quantum energy estimator that outputs clean states without phases or garbage, which sequence of techniques is employed to minimize the number of garbage registers while also eliminating phases, and what is the resulting query complexity bound assuming α is bounded away from 1?
1) First invoke Lemma 8 to eliminate garbage, then apply Lemma 7 to minimize garbage registers, resulting in query complexity O(α⁻¹ log(δ⁻¹) (2n + log(α⁻¹)))
2) First use Lemma 7 to minimize garbage registers, then Lemma 8 to eliminate phases, yielding query complexity O(nα log(δ) + log(α))
3) Apply only Lemma 8 to remove all phases, with query complexity O(α log(n) + δ)
4) Use Theorem 15 directly without any lemmas, resulting in query complexity O(2ⁿα log(δ⁻¹))
5) Alternate invoking Lemma 7 and Lemma 8 multiple times, leading to query complexity O(α⁻¹ n² log(δ))
6) Apply Lemma 7 to eliminate garbage and Lemma 8 for phase removal, obtaining query complexity O(n log(α) log(δ))
7) Discard all garbage after measurement without applying lemmas, giving query complexity O(α⁻¹ n log(log(δ)))
✓ Correct Answer:
The correct answer is 1) First invoke Lemma 8 to eliminate garbage, then apply Lemma 7 to minimize garbage registers, resulting in query complexity O(α⁻¹ log(δ⁻¹) (2n + log(α⁻¹))).
📚 Reference Text:
phases and garbage from Theorem 15 can be combined with Lemma 8 and Lemma 7 to make an energy estimator without phases and without garbage with query complexity bounded by: O/parenleftBig α−1log(δ−1)/parenleftBig 2n+ log/parenleftBig α−1/parenrightBig/parenrightBig/parenrightBig (173) Accepted in Quantum2021-10-14, click title to verify. Published under CC-BY 4.0. 33 assuming that αis bounded away from 1 by a constant. Furthermore, even when there is no rounding promise, there exists an algorithm that, given an eigenstate |ψj/angbracketrightof the Hamiltonian λj, for anyδ >0performs a transformation δ-close in diamond norm to the map: |ψj/angbracketright/angbracketleftψj|→/parenleftBig p|floor(2nλj)/angbracketright/angbracketleftfloor(2nλj)|+ (1−p)|λ/prime j/angbracketright/angbracketleftλ/prime j|/parenrightBig |ψj/angbracketright/angbracketleftψj|(174) wherepis some probability and λ/prime j=floor(2nλj)−1mod 2nis an erroneous estimate. Just as in Corollary 13, the performance is the same except that 0< α < 1can be any constant. Proof.As with Corollary 13, we write ηkto make the kdependence explicit and demand accuracyδamp,k= (1−10−mamp)(δ2−k−1)2/8for thek’th bit. From Lemma 14 we obtain an asymptotic upper bound r(t,ε)∈O/parenleftbigt+ log(ε−1)/parenrightbig. Again, recall from Lemma 11 that Mηk→δamp∈O/parenleftBig η−1 klog(δ−1 amp)/parenrightBig . The asymptotic query complexity of the iterative energy estimator from Theorem 15 is then bounded by: O/parenleftBig (ηk−δcos)−1log(δ−1 amp,k)·(2n−k+ log(δcos,k))/parenrightBig (175) ≤O/parenleftBig (1−10−mcos)−1η−1 klog((1−10−mamp)−122(k+1)δ−28)·(2n−k+ log(10mcosη−1 k))/parenrightBig (176) ≤O/parenleftBig η−1 klog(2k+1δ−1)·(2n−k+ log(η−1 k))/parenrightBig (177) Next we invoke Lemma 8 to remove the phases and the garbage, doubling the query complexity. We do this before invoking Lemma 7, because Lemma 7 involves blowing up the number of garbage registers by a factor of n. While we could also invoke Lemma 7 and then invoke Lemma 3 to obtain a map without garbage, this would involve many garbage registers sitting around waiting to be uncomputed for a long time. If we invoke Lemma 8 first we get rid of the garbage immediately. Finally, we invoke Lemma 7 to turn our iterative energy estimator without garbage and phases into a regular energy estimator without garbage and phases. As in Corollary 13, we observe that η0=α/2and fork >0we haveηkbounded from below by a constant. Then, the total query complexity is: O/parenleftBigg η−1 0log(20+1δ−1)/parenleftBig 2n−0+ log/parenleftBig η−1 0/parenrightBig/parenrightBig +n−1/summationdisplay k=1η−1 klog(2k+1δ−1)/parenleftBig 2n−k+ log/parenleftBig η−1 k/parenrightBig/parenrightBig/parenrightBigg (178) ≤O/parenleftBigg α−1log(δ−1)/parenleftBig 2n+ log/parenleftBig α−1/parenrightBig/parenrightBig +n−1/summationdisplay k=1(k+ log(δ−1))2n−k/parenrightBigg (179) ≤O/parenleftBig α−1log(δ−1)/parenleftBig 2n+ log/parenleftBig α−1/parenrightBig/parenrightBig + 2(2n−n−1) + (2n−2) log(δ−1)/parenrightBig (180) ≤O/parenleftBig α−1log(δ−1)/parenleftBig 2n+ log/parenleftBig α−1/parenrightBig/parenrightBig/parenrightBig (181) Next, we show that it is possible to implement a map that, given an eigenstate |φj/angbracketright, measures an estimate that is either floor (2nλj)or floor (2nλj)−1mod 2nwith some probability. Note that this is not the same algorithm as above. Just as with phase estimation, it is only the first bit that actually needs the rounding promise, and all other bits are guaranteed to be deterministic. Accepted in Quantum2021-10-14, click title to verify. Published under CC-BY 4.0. 34 The first bit performs a map of the form: |0n/angbracketright|0...0/angbracketright|ψj/angbracketright→/parenleftBig√p|0/angbracketright|gar0,j/angbracketright+/radicalbig 1−p|1/angbracketright|gar1,j/angbracketright/parenrightBig |ψj/angbracketright (182) We immediately see that Lemma 8 cannot be used to perform uncomputation here, be- cause uncomputation only works when p= 1orp= 0. Instead, we simply measure the output register containing |0/angbracketrightor|1/angbracketrightand discard the garbage. This would damage any superposition over
Question 122multiple-choice
In quantum information theory, the study of symmetries in multi-qubit systems often utilizes the representation theory of the symmetric group and matrix algebras constructed from its actions. Schur-Weyl duality provides a powerful framework for decomposing tensor product spaces and understanding their associated algebras.
Which statement accurately describes the decomposition of the Swap Matrix Algebra \( M_\text{swap}^n \) based on representation theory?
1) It decomposes into direct sums indexed by all possible partitions of \( n \).
2) It is equivalent to the full group algebra \( \mathbb{C}[S_n] \) without quotients.
3) Its decomposition is indexed by single-row Young diagrams corresponding to trivial representations.
4) The swap algebra is irreducible and cannot be decomposed further for any \( n \).
5) It decomposes as a direct sum over the irreducible representations of \( S_n \) corresponding to two-row Young diagrams.
6) Its structure is determined solely by the action of \( \mathrm{GL}_2(\mathbb{C}) \) on \( (\mathbb{C}^2)^{\otimes n} \).
7) The decomposition involves only the alternating representation of the symmetric group.
✓ Correct Answer:
The correct answer is 5) It decomposes as a direct sum over the irreducible representations of \( S_n \) corresponding to two-row Young diagrams..
📚 Reference Text:
by HG=/summationdisplay (i,j)∈E(G)2wij/parenleftig I−Swapij/parenrightig (2.14) Now, observe that we can write the right-hand side of Eq. (2.12) as /vextendsingle/vextendsingle/vextendsingleψ(i j)−1(1)/angbracketrightig ⊗···⊗/vextendsingle/vextendsingle/vextendsingleψ(i j)−1(i)/angbracketrightig ⊗/vextendsingle/vextendsingle/vextendsingleψ(i j)−1(j)/angbracketrightig ⊗···⊗/vextendsingle/vextendsingle/vextendsingleψ(i j)−1(n)/angbracketrightig , (2.15) where (ij)∈Snis the transposition of objects iandj. This is in fact a representation of Snon(C2)⊗ndefined by ρn(π) (|ψ1⟩⊗···⊗|ψn⟩) =/vextendsingle/vextendsingle/vextendsingleψπ−1(1)/angbracketrightig ⊗···⊗/vextendsingle/vextendsingle/vextendsingleψπ−1(n)/angbracketrightig (2.16) Definition 2.4. The algebra ρn(C[Sn]), for the representation/parenleftbigρn,(C2)⊗n/parenrightbigin Eq. (2.16) , is called the Swap Matrix Algebra Mswap n. Example 2.5. For smalln, the swap algebra Mswap n can be explicitly determined. Firstly, since the swap matrices Swapijare defined in terms of a representation of Sn, there is a ∗-homomorphism C[Sn]→Mswap n,(i j)∝⇕⊣√∫⊔≀→Swapij, (2.17) whenceMswap n is isomorphic to a semisimple quotient of the group algebra C[Sn] of the symmetric group Sn. Thus representation theory of Sncan be used to study Mswap n. For instance,S3has two one-dimensional representations (the trivial one and the signature), and one two-dimensional one, hence C[S3]∼=C⊕C⊕M2(C). Since dimMswap 3 = 5, this immediately yields Mswap 3∼=C⊕M2(C). Similarly, classifying irreducible representations for S4gives C[S4]∼=C⊕C⊕M2(C)⊕M3(C)⊕M3(C), which together with dim Mswap 4 = 14 implies that Mswap 4∼=C⊕M2(C)⊕M3(C). The general characterization of the Swap Matrix Algebra follows from Schur-Weyl duality [EGH+11] ofSnandGL2(C), the group of invertible 2×2complex matrices. The natural representation of GL2(C)on(C2)⊗nis defined by the diagonal action of the group elements on tensor products of |ψ1⟩,|ψ2⟩,...|ψn⟩∈C2, ζn(g) (|ψ1⟩⊗|ψ2⟩⊗···⊗|ψn⟩) =g|ψ1⟩⊗g|ψ2⟩⊗···⊗g|ψn⟩, g∈GL2(C).(2.18) The irreducible modules of GL2(C)are indexed by two row Young diagrams with an unbounded number of boxes. We denote the former by L[n−k,k]forn∈Nandk= 0,...,⌊n 2⌋. In fact,L[n−k,k]is the space of all linear maps from V[n−k,k]to(C2)⊗nthat commute with the action of Sn, that is L[n−k,k]= HomSn(V[n−k,k],(C2)⊗n). (2.19) The following lemma is essentially a restatement of Schur-Weyl duality for SnandGL2(C). Accepted in Quantum2024-03-25, click title to verify. Published under CC-BY 4.0. 12 Lemma 2.6. The algebras Mswap n andζn(C[GL 2(C)])are centralizers of each other inside End((C2)⊗n) =M2n(C). Moreover, the space (C2)⊗ndecomposes under the action of the direct product GL2(C)×Snas (C2)⊗n∼=⌊n 2⌋/circleplusdisplay k=0L[n−k,k]⊗V[n−k,k]. (2.20) Proof. See, e.g., [EGH+11, Sec. 5.19] or [Pro07, §6.1]. Since the action of SnonL[n−k,k]is trivial, as an Sn-module, the space (C2)⊗ndecom- poses by Lemma 2.6 into irreducible modules V[n−k,k]with multiplicities as follows: (C2)⊗n=⌊n 2⌋/circleplusdisplay k=0(V[n−k,k])dim(L[n−k,k]). (2.21) The Weyl character formula [EGH+11, Theorem 5.22.1] gives an explicit formula for dim(L[n−k,k]). Further, Lemma 2.6 immediately leads to the following characterization of Mswap n in terms of the irreducible representations of C[Sn]; cf. [Pro21, (1.12)]. Theorem 2.7. The Swap Matrix Algebra decomposes into the direct sum of simple algebras generated by the two row irreps of the symmetric group. That is, we have Mswap n∼=⌊n 2⌋/circleplusdisplay k=0ρ[n−k,k](C[Sn]). (2.22) Proof. This is immediate from Lemma 2.6 and Eq. (2.21). 2.4 Quantum Max Cut and Irreps Now we discuss how the decomposition of the Swap Matrix Algebra into irreps described in the previous section can be applied to calculations related to the eigenvalues of Quantum Max Cut Hamiltonians. Among other benefits the calculations in this section are essential to Section 6. Definition 2.8. Letλ⊢nbe any partition labeling an irrep of Sn, and letGbe
Question 123multiple-choice
In descriptive data mining, subgroup discovery aims to identify meaningful subpopulations within complex networks, often using graph-based analysis techniques. Advanced methods seek to overcome the limitations of traditional approaches by incorporating behavioral interactions between nodes.
Which methodological innovation most effectively addresses the challenge of uncovering indirect behavioral influences between nodes in non-uniform, user-generated datasets?
1) Relying exclusively on explicit graph connections between nodes
2) Applying uniform weighting to all node interactions regardless of behavior
3) Ignoring implicit relationships during subgroup classification
4) Using unweighted digraphs for subgroup analysis
5) Performing subgroup division based only on node degree centrality
6) Forming a weighted complete digraph that incorporates implicit behavioral interactions for subgroup discovery
7) Selecting subgroups based solely on demographic attributes
✓ Correct Answer:
The correct answer is 6) Forming a weighted complete digraph that incorporates implicit behavioral interactions for subgroup discovery.
📚 Reference Text:
Title: Subgroup Discovery Method Based on User Behavior Analysis Year: 2018 Paper ID: accb2822a62d4e8829dbe4d0bd8a77da3f0880c8 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/accb2822a62d4e8829dbe4d0bd8a77da3f0880c8 Abstract: Subgroup discovery and analytics is an important tool for descriptive data mining. Most methods about subgroup discovery are based on the graph structure, ignoring the hidden mutual influence between nodes. In this paper, we consider the above issues, re-think the impact of implicit relationships between nodes, base on the network structure partition method, make the interaction behavior between user nodes reflected in the graph structure. A method of subgroup discovery based on behavior interaction (abbreviated as SDBI) is proposed. First, extract some features of data by using the method named "Initial classification of prominent data ", and then extract the remaining data in order according to the importance of the data, and construct the selected data node as a weighted complete digraph. Afterwards, the SDBI were used to divide subgroups. The results prove that the proposed method can effectively consider the problem of data non-uniformity and accurately divide the results.
Question 124multiple-choice
In quantum computing, iterative estimators are crucial for algorithms that extract binary representations of eigenvalues, such as phase and energy estimation. Managing computational byproducts, often called garbage, and controlling error accumulation are essential for practical and scalable quantum algorithm design.
Which statement best captures how the total error of a non-iterative estimator constructed by stitching together multiple coherent iterative estimators is controlled in such modular quantum algorithms?
1) The total error is the product of individual estimation errors at each bit.
2) The total error is equal to the largest error among all bits estimated.
3) The total error accumulates linearly with the number of bits extracted.
4) The total error is bounded by δ, leveraging the triangle inequality of the diamond norm across all bits.
5) The total error is always zero when garbage is uncomputed at the end of the process.
6) The total error is independent of the rounding promise used in the protocol.
7) The total error increases exponentially with the number of least significant bits estimated.
✓ Correct Answer:
The correct answer is 4) The total error is bounded by δ, leveraging the triangle inequality of the diamond norm across all bits..
📚 Reference Text:
idea into the modular framework of Definition 2 and Lemma 3. A ‘coherent iterative estimator’ obtains a single bit of the estimate, given access to all the previousbits. Severalinvocations of acoherent iterative estimator yielda regularestimator as in Definition 2. Furthermore, we can choose to uncompute the garbage at the very end using Lemma 3, or, as we will show, we can remove the garbage early, which prevents it from piling up. Definition 6. Acoherent iterative phase estimator is a protocol that, given some n, α, anyk∈{0,...,n−1}, and any error target δ>0, produces a quantum circuit involving calls to controlled- UandU†. If the unitary Usatisfies an (n,α)-rounding promise, then this circuit implements a quantum channel that is δ-close to some map that performs: |0/angbracketright|∆k/angbracketright|ψj/angbracketright→|bitk(λj)/angbracketright|∆k/angbracketright|ψj/angbracketright (74) Here bitk(λj)is the (k+ 1)’th least significant bit of an n-bit binary expansion, and |∆k/angbracketright is ak-qubit register encoding the kleast significant bits. (For example, if n= 4and λj= 0.1011...then bit 2(λj) = 0and∆2= 11.) Note that the target map is only constrained on a subspace of the input Hilbert space, and can be anything else on the rest. Ancoherent iterative energy estimator is the same thing, just with controlled- UH andU† Hqueries to some block-encoding UHof a Hamiltonian Hthat satisfies an (n,α)- rounding promise. Accepted in Quantum2021-10-14, click title to verify. Published under CC-BY 4.0. 16 A coherent iterative estimator can also be with garbage and/or with phases, just like in Definition 2. Lemma 7. Stitching together coherent iterative estimators. Given a coherent iterative phase/energy estimator with query complexity Q(n,k/prime,α,δ/prime), we can construct a non-iterative phase/energy estimator with query complexity: n−1/summationdisplay k=0Q/parenleftBig n,k,α,δ·2−k−1/parenrightBig (75) The non-iterative estimator has phases if and only if the coherent iterative estimator has phases, and if the coherent iterative estimator has mqubits of garbage then the iterative estimator has nmqubits of garbage. Proof.We will combine nmany coherent iterative phase/energy estimators, for k= 0,1,2,..,n−1. The diamond norm satisfies a triangle inequality so if we let the k’th iterative estimator have an error δk:=δ2−k−1then the overall error will be: n−1/summationdisplay k=0δk≤δ 2n−1/summationdisplay k=02−k=δ 2(2−21−n)≤δ (76) So now all that is left is to observe that the exact iterative estimators chain together correctly. This should be clear by observing that for all k>0: |∆k/angbracketright=|bitk−1(λj)/angbracketright⊗...⊗|bit0(λj)/angbracketright (77) and|∆0/angbracketright= 1∈Csince when k= 0there are no less significant bits. So the k’th iterative estimator takes the kleast significant bits as input and computes one more bit, until finally atk=n−1we have: |bitn−1(λj)/angbracketright|∆n−1/angbracketright=|bitn−1(λj)/angbracketright⊗...⊗|bit0(λj)/angbracketright=|floor(2nλj)/angbracketright(78) The total query complexity is just the sum of the ninvocations of the iterative estimators fromk= 0,...,n−1with errorδk. If the iterative estimator has garbage, then the garbage from each of the ninvocations just piles up. Similarly, if the estimator is with phases, and has an eiϕj,kfor thek’th invocation, then the composition of the maps will have a phase/producttextn−1 k=0eiϕj,k. Lemma 8. Removing garbage and phases from iterative estimators. Given a coherent iterative phase/energy estimator with phases and/or garbage and with query com- plexityQ(n,k,α,δ/prime)that has a unitary implementation, we can construct a coherent it- erative phase/energy estimator
Question 125multiple-choice
In the study of algebraic structures and their associated matrix theory, diagonalizability over localizations of the integers and the bijectivity of natural maps play critical roles in arithmetic geometry and the theory of moduli spaces. Special attention is often given to properties of symmetric matrices and the impact of localization at specific primes.
Which statement best describes why symmetric matrices over the localization Zₗ may fail to be Zₗ-diagonalizable specifically at the prime l = 2, and how this issue can be resolved?
1) Because the determinant of any symmetric matrix over Z₂ is always zero, diagonalization is impossible.
2) Since invertible matrices do not exist over Z₂, symmetric matrices cannot be diagonalized.
3) The failure arises because all entries become units at l = 2, making diagonalization trivial.
4) Diagonalizability fails at l = 2 due to the presence of nontrivial zero divisors that obstruct invertibility.
5) At l = 2, only skew-symmetric matrices are diagonalizable, not symmetric ones.
6) Symmetric matrices over Z₂ may not be diagonalizable, but adding diagonal blocks with appropriate powers of 2 allows diagonalization.
7) The obstruction at l = 2 is due to the lack of a well-defined trace, which prevents diagonalization.
✓ Correct Answer:
The correct answer is 6) Symmetric matrices over Z₂ may not be diagonalizable, but adding diagonal blocks with appropriate powers of 2 allows diagonalization..
📚 Reference Text:
the definition of the maps. The last assertions fol- low from Propositions III. 1.1 and III. 1.3. Now we proceed to show that ψ is bijective. Let A be a subring of the reals, R. We say that a g x g matrix si over R is Λ-diagonalizable if there exists an invertible matrix SQ over A such that SQS/SI is diagonal. Let Z£ denote the localization (not the completion) of Z at the prime £. THEOREM III.2.2. Let ati = ord. siio and si be the matrix (ais). Suppose that si is Z£-diagonalίzable for every prime £. Then the map φ: Gg/Γ —> Ak is bijective (for the matrix (st/tj)). Proof. Let So be an invertible matrix over Z£ diagonalizing si and let To = So'1. Replacing So and To by integer multiples prime to £ we get matrices S and T over Z with ST — nl, in, £) = 1 and Ss/St diago- nal. Let btj be defined as in §111.1. Then the matrix (ord. btj) which is equal to SAkχ (Eλ)k x ... x (#,)* is bijective. Therefore φ is bijective too. The following simple result will be proved in the appendix. LEMMA. Let (ai3) be a symmetric matrix ivith entries in Z£. Then: 1) if 6 Φ 2, (atJ) is Zr diagonalizable. 2) if £ — 2, there exist integers m19 , ms which are powers of 2 such that (ai3) 0 diag. (m19 , ms) is Zrdiagonal- ίzable. Let (sfij) be our matrix satisfying the Riemann conditions. Combin- ing the above lemma with Theorem III.2.3 we
Question 126multiple-choice
Quantum computing can utilize qudits, which are quantum systems with more than two levels, to enhance computational efficiency and scalability. Universal gate sets for qudits are essential for implementing arbitrary quantum algorithms in higher-dimensional systems.
Which combination of gates forms a universal gate set for qutrits (three-level quantum systems), enabling the dense generation of SU(3) and the qutrit Clifford group?
1) CNOT, H, T
2) SUM₃, H₃, and gates from {P[0]₃, P[1]₃, P[2]₃}
3) Toffoli, Pauli-X, and SWAP
4) Fredkin, Hadamard, and π/8
5) Controlled-Z, Phase, and SUM₂
6) SUM₄, H₄, and P[3]₄
7) Pauli-Y, Hadamard, and Controlled-SUM
✓ Correct Answer:
The correct answer is 2) SUM₃, H₃, and gates from {P[0]₃, P[1]₃, P[2]₃}.
📚 Reference Text:
2 2N2). So the qudit method has a (log2d)2 scaling advantage over the qubit case. Furthermore, in this FIGURE 1 | The schematic circuit of Cm[Rd]with C2[Rd]and C2[Pd(p,q)]. The horizontal lines represent qudits. The auxiliary qudits initialized to |0〉is denoted by the red lines and the black lines denoting m controlling qudits. The two-qudit controlled gates is shown as theverticle lines. P d(p,q)is the permutation of |p〉and |q〉state, and Rdis either X(l) dorZd. Frontiers in Physics | www.frontiersin.org November 2020 | Volume 8 | Article 589504 3Wang et al. Qudits and High-Dimensional Quantum Computing reviewed method, for an arbitrary unitary U∈SU(N), from Eqs 7 and 8 Neigenoperators is needed and each can be decomposed with three rotations shown in Eq. 9 . Deriving from the appendix of Ref. 108;Uj,Ncan be decomposed with less than 3 dn−1multiple controlled operations. Finally, as Figure 1 has shown, Cm[Rd] needs mnumber of C2[Rd]andC2[Pd(p,q)].Udcan be composed with d−1 numbers of X(l) das in Eq. 3 . Therefore the total number of primitive operations Lin this decomposition method is L≤2N×3dn−1×n×(d−1)+N×n≤6nd2n+ndn. (12) It is clear that there is an extra factor of nreduction in the gate requirement as the number scale of this method is O(nN2). The other advantage is these primitive qudit gates can be easily implemented with fewer free parameters [ 108]. For qudit quantum computing, depending on the implementation platform, other universal quantum gate setscan be considered. For example, in a recent proposal fortopological quantum computing with metaplectic anyons, Cuiand Wang prove a universal gates set for qutrit and qupit systems, for a qupit being a qudit with pdimensions and pis an prime number larger than 3 [ 38]. The proposed universal set is a qudit analogy of the qubit universal set and it consists severalgeneralized qudit gates from the universal qubit set. The generalized Hadamard gate for qudits H dis Hd⎥vextendsingle⎥vextendsingle⎥vextendsingle⎥vextendsinglej〉/equals1⎥radicaltpext⎥radicaltpext d√⎥summationdisplayd−1 i/equals0ωij|i〉,j∈{0,1,2,... ,d−1}, (13) where ω:/equalse2πi/d .(14) The SUM dgate serves as a natural generalization of the CNOT gate SUM d|i,j〉/equals|i,i+j(modd)〉,i,j∈{0,1,2,... ,d−1}.(15) The Pauli σz, with the π/8 gate as its 4th root, can be generalized to Q[i]gates for qudits, Q[i]d⎥vextendsingle⎥vextendsingle⎥vextendsingle⎥vextendsinglej〉/equalsωδij⎥vextendsingle⎥vextendsingle⎥vextendsingle⎥vextendsinglej〉, (16) withωdefined by Eq. 14 and the related P[i]gates are P[i]d⎥vextendsingle⎥vextendsingle⎥vextendsingle⎥vextendsinglej〉/equals⎥parenleftbig−ω2⎥parenrightbigδij⎥vextendsingle⎥vextendsingle⎥vextendsingle⎥vextendsinglej〉,i,j∈{0,1,2,... ,d−1}. (17) In general Q[i]pis always a power of P[i]pifpis an odd prime. The proposed gate set for the qutrit system is the sum gate SUM 3, the Hadamard gate H3and any gate from the set{P[0]3,P[1]3,P[2]3}.A sa na n a l o g u eo ft h es t a n d a r d universal set for qubit {CNOT ,H,T/equalsπ/8−gate}, the qutrit set generate the qutrit Cliffo rd group whereas the qubit set generate the qubit Clifford group (the de finition of the Clifford group can be found in Section 2.2.1 ). Whereas the rigorous proof can be found in Ref. 38; the proving process follows the idea introduced in Ref. 23that the gate SUM 3is imprimitive, and the Hadamard H3and any gate from {P[0]3,P[1]3,P[2]3} generates a dense subgroup of SU(3). Similarly, the proposed gate set for the qupit
Question 127multiple-choice
Quantum algorithms for the Hidden Subgroup Problem (HSP) over real vector spaces rely on lattice theory, efficient state construction, and sampling techniques. Understanding how these components interact is essential for grasping the algorithmic framework in continuous group settings.
Which of the following is a critical step in quantum algorithms for solving the Hidden Subgroup Problem over R^m that ensures efficient extraction of the hidden subgroup information?
1) Sampling an approximation to a random point in the reciprocal lattice L* using a Lipschitz function with compact support
2) Encoding the hidden subgroup directly as a classical bitstring prior to quantum state preparation
3) Restricting the group structure exclusively to finite Abelian groups without embedding discrete groups into continuous spaces
4) Avoiding the use of bounded condition numbers when analyzing reduced lattice bases
5) Constructing quantum states solely from discrete periodic functions without generalizing to continuous functions
6) Applying measurement exclusively in the standard basis rather than the Fourier basis
7) Prohibiting any parameter scaling to preserve original lattice geometry during embedding
✓ Correct Answer:
The correct answer is 1) Sampling an approximation to a random point in the reciprocal lattice L* using a Lipschitz function with compact support.
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improved analysis of [BK93] in Section 4.3. It can be shown that the condition number of a reduced basis is bounded so that the dual lattice, which is the hidden subgroup, can be computed. These provide the main ingredients in the full proof. Theorem 6.1. There is a polynomial time quantum algo- rithm for solving the HSP over Rm. In Section 6.2 we outline the steps of the algorithm and indicate how to derive the probability expression. First we show that the known cases of the Abelian HSP can be re- duced to the new continuous case in Definition 1.1 over the HSP instance Rm. 6.1 Application: Reduction to G=Rm Our HSP algorithm is applicable to Abelian groups of the formRk×Zl×(R/Z)s×H, whereHis finite. We call such groups “elementary”. The reduction to G=Rk×Zlis straightforward. In the case of interest, the hidden subgroup Lis a full-rank lattice in G⊆Rk×Rlsuch thatλ1(L∩Rk)/greaterorequalslant λandd(L)/lessorequalslantdfor some fixed numbers λandd. We now describe the further reduction to the group ˜G=Rk+l. The main idea can be illustrated in the one-dimensional case, where the parameter λhas no meaning. We embed G=Zinto ˜G=Rin the standard way, set ν= 2−qfor someq/greaterorequalslant2, and define the R-oraclegin terms of the Z- oraclefas follows: ˛˛g(x)¸ =c0˛˛strν(t)¸ ⊗˛˛f(s)¸ + c1˛˛strν(t−1)¸ ⊗˛˛f(s+ 1)¸ , (6.2) wheres=⌊x⌋, t =x−s, c 0= cos`π2t´ , c1= sin`π 2t´ . It is clear that gis a continuous function. If fis a periodic function, then gis also periodic with the same period. To construct the state |g(x)/angbracketright using the original oracle f, we compute sandt, use them as parameters in the following sequence of operations, and “uncompute” sandt: |0/angbracketright /mapsto→X zcz|z/angbracketright /mapsto→X zcz|z/angbracketright ⊗˛˛f(s+z)¸ /mapsto→X zcz˛˛strν(t−z)¸ ⊗˛˛f(s+z)¸ , wherez∈ {0, 1}. The last step, |z/angbracketright /mapsto→ | strν(t−z)/angbracketright, re- quires that we discriminate between the states |strν(t)/angbracketrightand |strν(t−1)/angbracketright. This is easy because the supports of those states on the ν-grid do not overlap. Let us now consider the general case, G=Rk×Zl. The groupGis embedded in ˜G=Rk+lby scaling the Zfactors byλ. This is to guarantee that λ1(˜L)/greaterorequalslantλ, where ˜Lis the image ofLunder the embedding. The other condition on the new hidden subgroup reads: d(˜L)/lessorequalslant˜d, where ˜d=dλl. The generalization of Eq. (6.2) is straightforward: ˛˛g(x,x 1,...,x l)¸ =X z1,...,z l∈{0,1} lO j=1|ψ(xj,zj)/angbracketright! ⊗˛˛f(x, s(x 1,z1),...,s(x l,zl)¸ ,(6.3) 300 wheres(x,z ) =⌊x/λ⌋ +z,|ψ(x,z)/angbracketright= cos`π 2t´˛˛strν(t)¸ , witht=x/λ−s(x,z ). Note that the terms in the above sum are mutually or- thogonal vectors. It can be shown that ghas parameters ˜a2=a2+l`π 2νλ(1 +ν)´2, ˜r2=r2+l(2νλ)2and/epsilon1. 6.2 An HSP algorithm for the group Rm Letfbe an (a,r,ε) oracle function for some full-rank lat- ticeL⊆Rmsuch thatλ1(L)/greaterorequalslantλandd(L)/lessorequalslantd(see Def- inition 1.1). The core part of our algorithm is a sampling subroutine that generates an approximation to a random point of the reciprocal lattice L∗. It works under certain assumptions about the oracle parameters. Letω:R→Cbe some Lipschitz function with unit L2- norm supported on the interval [0, 1]. For example, ω(x) =(√2 sin(πx) for x∈[0,1], 0 otherwise.(6.4) Let us also choose a sufficiently large number ∆ = 2q1and a sufficiently small number
Question 128multiple-choice
Representation theory of symmetric groups and invariant theory play a central role in quantum information, combinatorics, and mathematical physics, especially regarding algebras constructed from group actions and their decompositions. Understanding the structure and labeling of projectors within such algebras requires detailed knowledge of Young diagrams and combinatorial coefficients.
In the algebra A(m,n), which is a subalgebra of the group algebra of the symmetric group Sm+n invariant under the adjoint action of Sm×Sn, the projectors in its center are explicitly labeled by which mathematical objects, and what key combinatorial property must these labels possess?
1) Pairs of partitions of m and n with coprime sizes
2) Single symmetric functions with nonzero Kronecker coefficients
3) Unordered sets of conjugacy classes in Sm+n with maximal cycle length
4) Pairs of Young diagrams with equal number of boxes and zero Littlewood-Richardson coefficient
5) Triples of irreducible representations with matching dimension
6) Young diagram triples (R, R1, R2) with nonzero Littlewood-Richardson coefficients
7) Necklaces corresponding to the minimal number of orbits under Sm×Sn action
✓ Correct Answer:
The correct answer is 6) Young diagram triples (R, R1, R2) with nonzero Littlewood-Richardson coefficients.
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generalisations of this task. For example Bob could send Alice a pair of Q-states, both with the same (R1,R2,R3)and with Kronecker coefficient 1or greater than 1, and Alice could be tasked with determining the triple of Young diagrams and determining whether the two states sent by Bob are linearly independent. If they are, then this would show that the Kronecker coefficient for the triple is greater than 1. This would require using QPE to determine the eigenvalues of elements in K(n)which are not necessarily central. The use of non-central elements of a permutation centraliser algebra to construct distinct multiplicity states (τ1,τ2) has been used [ 37] in the context of multi-matrix invariants (related to the algebra A(m,n) discussed in section 7). Holography. Following the similarity between the projector detection in this section and the detection in Z(C(Sn))from section 3 it is natural to ask if we can formulate a holographically dual gravitational detection problem for K(n). This algebra has been studied [ 30,31] in connection with invariant observables for tensor models. These in turn have been related [ 70] to the SYK models [ 71,72]. A holographic correspondence between SYK and near- AdS 2quantum gravity has been discussed (see the original papers [ 73–75] and a recent review which discusses the correspondence and associated subtleties [ 76]). It is an interesting question whether near-AdS2 gravity allows a classical gravity dual of the projector detection in K(n)analogous to the classical gravity dual of projector detection in Z(C(Sn))based on LLM geometries [ 11] which we described in section 4. 7 Detection of the Littlewood-Richardson projector Given two non negative integers mandn, there is a PCA A(m,n)which is relevant for the counting of polynomial functions of two matrices X,Yinvariant under adjoint action by unitary matrices U: (X,Y )→(UXU†,UYU†). We present a review of the background from [30]. A(m,n)is the subalgebra of C(Sm+n)made of the elements that are invariant under the adjoint action of Sm×Sn. There is a basis of A(m,n)labelled by multi-necklaces with mbeads of one colour and nbeads of another colour. These are in 1-1 correspondence with orbits in Sm+ngenerated by the conjugation action of γ∈Sm×Sn⊂Sm+n. Withr running over these necklaces, Er=/summationdisplay γ∈Sm×Snγσ(r)γ−1, (7.1) whereσ(r)∈Sm+nis a representative of the r’th orbit. One shows that A(m,n)is a semi-simple associative algebra and therefore admits a Wedderburn-Artin decomposition. It is proved that Dim(A(m,n)) =/summationdisplay R/turnstileleftm+n,R1/turnstileleftm,R 2/turnstileleftng(R1,R2,R)2(7.2) whereg(R,R 1,R2)is the so-called Littlewood-Richardson (LR) coefficient. There is a Wedderburn-Artin basis of A(m,n)in the form QR R1,R2;µνwithR,R 1,R2 being Young diagrams with m+n,m,nboxes respectively. The Littlewood-Richardson – 28 – JHEP05(2023)191coefficient determines the range of the indices µ,ν:1≤µ,ν≤g(R1,R2,R). Explicit formulae for QR R1,R2;µνare known in terms of matrix elements of permutations in the irrepRofSm+nalong with branching coefficients for the reduction of the irrep Rinto representations of the subgroup Sm×Sn. The centreZ(A(m,n))is spanned by projectors labelled by triples (R1,R2,R3)with non-vanishing g(R1,R2,R). They can be written in terms of characters: PR R1,R2=dRdR1dR2 (m+n)!m!n!/summationdisplay σ∈Sm+n/summationdisplay σ1∈Sm/summationdisplay σ2∈SnχR(σ)χR1(σ1)χR2(σ2)σ(σ1◦σ2)(7.3) TASK: Identify a triple (R,R 1,R2)for a LR-projector PR R1,R2.We proceed again by making explicit
Question 129multiple-choice
Isogeny-based cryptography leverages complex mathematical structures, such as quaternion algebras and maximal orders, to achieve security against classical and quantum attacks. Understanding how algorithmic techniques interact with parameters like powersmoothness is crucial for evaluating protocol security.
Which algorithmic procedure is specifically designed to find an element in the maximal order O₀ of a quaternion algebra with a prescribed norm M, supporting the construction of isogenies with desired properties?
1) Cornacchia's algorithm
2) StrongApproximationF(N, µ₀)
3) KLTPM
4) Quantum Decomposition Algorithm
5) RepresentInteger′ R+Rj(N, A, B)
6) Lattice Reduction
7) RepresentIntegerO₀
✓ Correct Answer:
The correct answer is 7) RepresentIntegerO₀.
📚 Reference Text:
As a direct consequence, th is breaks pSIDH quantumly since Nis a large prime in pSIDH. As mentioned previously, the IsERP is easy when Nis powersmooth. We briefly discuss some approaches to solve the general case in Appendix D. Title Suppressed Due to Excessive Length 21 In Section 5.1, we give a summary of useful known algorithms and pro vide variantsfor RepresentInteger ,StrongApproximation andKLPTtobetteraccommo- date our specific application. Following this, we introduce our primary strategy for resolving the PQLP in Section 5.2. In Section 5.3, we deal with a crit ical technical point which we introduce as the Quaternion Decomposition problem. The crux of this problem, and indeed our main conceptual contribut ion, is the decomposition of σ0into elements that are easy to lift, and elements already possessing a powersmooth norm. Finally in Section 5.4, we provide a qu antum algorithm that solves the PQLP. 5.1 Algorithmic building blocks Our algorithm for the PQLP is founded on algorithmic building blocks initia lly introduced in [ KLPT14 ] and later extended in other work,such as [ FKL+20]. We provide a brief recap of these algorithms here, along with several n ew variants tailored to suit our requirements. We fix logcpto be our powersmooth bound for some constant c, and this bound is inherently implied whenever we reference the term ‘powersmooth’. As in [KLPT14 ], for eachp, we fix a special p-extremal maximal order O0. O0=/braceleftBigg Z/an}bracke⊔le{⊔i,1+j 2/an}bracke⊔ri}h⊔wherei2=−1,j2=−p, forp≡3 mod 4, Z/an}bracke⊔le{⊔1+i 2,j,ci+k q/an}bracke⊔ri}h⊔wherei2=−q,j2=−p,forp≡1 mod 4,(3) wherecis any root of x2+pmodq. In the second case, qis required to satisfy thatq≡3 mod 4 is a prime and/parenleftbig−p q/parenrightbig = 1. We add one extra condition that (q,N) = 1. Under the generalized Riemann hypothesis (GRH), the smallest qis of sizeO(log2p). Foreach O0, weidentify asuborderofthe form R+RjforR=Z[i] (note that wearemakingaslightlydifferent choiceherethan the Rin[KLPT14 ] wherethey always take Rto the maximal order in Q(i)). For an element α=a+bi∈R, we useReR(α) to denote aandImR(α) to denote b. LetDdenote the index [O0:R+Rj], then D=/braceleftBigg 4,forp≡3 mod 4, 4q,forp≡1 mod 4.(4) We will now detail the algorithmic building blocks, sourced from [ KLPT14 ] or [FKL+20]. –Cornacchia (M): on an input M∈Z, outputs either ⊥ifMcannot be repre- sented by a fixed quadratic form f(x,y), or a solution x,yto the equation M=f(x,y). –RepresentIntegerO0(M): on an input M > p, outputsγ∈ O0such that n(γ) =M. –StrongApproximationF(N,µ0): on inputs an integer F > pN4, a primeN andµ0∈Rj, outputsλ /∈NZandµ∈ O0of norm dividing Fsuch that µ=λµ0modNO0. 22 M. Chen, M. Imran, G.Ivanyos, P.Kutas, A.Leroux, C. Petit –KLTPM(I): on inputs an integer M >p3and an ideal I, outputs an equiva- lent idealJsuch thatn(J) =M. Let us denote the output γofRepresentIntegerO0(M) asC+DjwithC,D∈ R. To fit our algorithm’s specific use case, we require not only that n(γ) is powersmooth, but also that C,Dsatisfy additional conditions relative to some inputsA,B∈R, which are determined by σ0from the PQLP. As a result, we introduce the following variant named RepresentInteger′ R+Rj(N,A,B). This variantnecessitatesmorerandomizedsteps to find the desiredou tputsC,D∈R. Algorithm 1: RepresentInteger′ R+Rj(N,A,B) Input:An integer NandA,B∈R Output: C,D∈Rsuch that: i)
Question 130multiple-choice
In knot theory, various methods are used to represent and classify links, including the use of braids and their closures. Certain theorems guarantee that all links can be represented using specific types of braid closures, which are foundational in the study of link invariants.
Which theorem asserts that every link can be represented as the closure of a braid using trace closure?
1) Birman's theorem
2) Alexander's theorem
3) Markov's theorem
4) Hilden's theorem
5) Jones's theorem
6) Artin's theorem
7) Reidemeister's theorem
✓ Correct Answer:
The correct answer is 2) Alexander's theorem.
📚 Reference Text:
we say that they are ambient isotopic . Alink invariant is association of links to algebraic objects (numbers, grou ps, modules, etc.) that respects the equivalence relation of ambient isotopy: equivalent li nks must be associated with the same object. One well-studied framework for studying links is to represent t hem as certain canonical “closures” of braids and explore the algebraic properties of braids that preserve li nk equivalence in this representation. We consider two such closures here: trace closure and plat closure. 4.1.1 Trace closure Trace closure of a braid is obtained by joining the ends (top t o bottom) as shown in Figure 2a; Alexander’s theorem [ 4] asserts that any link can be obtained in this fashion. Of cou rse, a given link can be represented as the closure of many different braids. It has been shown by Markov [ 25] that braids that yield equivalent knots under the trace clo sure are related by the Markov moves . These moves are θη←→ηθ, θ,η ∈Bn,and θ←→θσ±1 n−1, θ ∈Bn−1⊂Bn. AMarkov trace is a mapφ:Bn→Cthat is well-behaved with respect to these moves: φ(θη) =φ(ηθ), φ(θσn−1) =zφ(θ), φ(θσ−1 n−1) = ¯zφ(θ), 12 braid (a) The trace closure.braid (b) The plat closure. Figure 2: Braid closures. wherez=φ(σn−1) and ¯z=φ(σ−1 n−1). With such a trace map, one can construct a link invariant by defining, for any braid θthat realizes the link, the quantity L(θ) = (z¯z)−(n−1)/2/parenleftbigg¯z z/parenrightbigge(θ)/2 φ(θ), (4.1) where the linking number, e(θ), is the sum of the exponents of the generators σiappearing in θ. It was shown by Tsohantjis and Gould [ 34] that the quantum doubles of finite groups yield link invaria nts by this approach. Specifically, let ( ρ,h) be an irrep of D(G) acting on the Hilbert space Vand letτdenote the natural representation of the braid group Bn, described in Section 2.2above, on the space V⊗n. Then the quantity Lρ,h(θ) = (dρ,h)−1/a\}bracketle{th/a\}bracketri}ht−e(θ) ρ Tr(τ(θ)) (4.2) is a link invariant, where /a\}bracketle{th/a\}bracketri}htρ=χρ(h)/dρandχρ,dρare the character and dimension of the irrep ρofZ(h). 4.1.2 Plat closure A related method to represent links by braids is the plat closure . This is defined on braids with an even number of strands where one takes pairs of adjacent strands o n the top (and bottom) and joins them as in Figure 2b. Similar to trace closure, it can be shown that any link can be represented as the plat closure of some braid. Birman [ 7] proved an analogue of Markov’s theorem for the plat closure . Theorem 1 (Birman [ 7]).Given two braids in B2nwith the same plat closure, there exists a sequence of moves from one to the other such that each move is one of the fol lowing: •(Double coset move) θ←→h1θh2, whereh1,h2∈H2n(defined below) and, •(Stabilization move) θ←→σ2nθ. Here,H2ndenotes the Hilden group , the subgroup of the braid group B2ngenerated by σ1, σ 2σ2 1σ2,andσ2iσ2i−1σ2i+1σ2i,∀i∈{1,...,n−1}. The quantum double of a finite group Glikewise yields link invariants by this approach. Consider the 2n-fold tensor power of an irrep Λ for
Question 131multiple-choice
In quantum photonics, linear optical unitaries are implemented using beamsplitters and phase shifters, but achieving universality requires extending the gate set. Universality enables the approximation of any unitary operation on the system's Hilbert space.
Which of the following modifications to a linear optics gate set is sufficient to achieve universality in quantum photonic circuits for m ≥ 3 modes?
1) Adding a second beamsplitter with a variable angle
2) Adding a single SNAP gate at a fixed mode, angle, and photon number
3) Introducing photon number-resolving detectors only
4) Including only additional phase shifters
5) Restricting to passive linear optics without ancilla photons
6) Using only displacement operations on each mode
7) Applying a controlled-Z gate between two modes
✓ Correct Answer:
The correct answer is 2) Adding a single SNAP gate at a fixed mode, angle, and photon number.
📚 Reference Text:
and universality Since Un,mis not universal in U(H), we instead ask whether it is a t-design for sufficiently large t. We first observe that the linear optical unitaries form a 1-design. This result is quite straightforward and is surely known; however, as we could not find an explicit statement in the literature, we provide the proof in Appendix B 4. Proposition 4. Letn, m≥2. The group Un,mof linear optical unitaries is a 1-design in U(H). However, Un,mis not a t-design for t≥2. This may be proven directly using representation theory, but it is also a trivial consequence of Theorem 7: Corollary 1. Letn, m≥2. Let Gbe a closed subgroup of U(H)extending the linear optical unitaries, Un,m⊆G⊆ U(H). IfGis a2-design in U(H), then Gis universal. In particular, since Un,mis not universal, it is not a t-design for t≥2. As a result of Corollary 1, if we want a 2-design extending Un,m, we are forced to work with a universal set. The natural question is whether we can at least obtain a nice generating set for U(H). Oszmaniec and Zimborás [8] showed that linear optics augmented by nearly any additional gate is sufficient to densely generate U(H). We further discuss this result in Appendix B 5. Theorem 8 ([8]).Consider the group Un,mof linear optical unitaries for n≥2photons in m≥2modes. Let V∈U(H) be any unitary gate with V̸∈Un,m. 1. For m > 2, the group generated by Un,mandVis universal. 2. For m= 2, define |ζ⟩=Pn a=0(−1)a|a, n−a⟩ ⊗ |n−a, a⟩.If [V⊗V,|ζ⟩⟨ζ|]̸= 0, (28) then the group generated by Un,mandVis universal. In Appendix B 6, we prove that the extension of linear optics by any single SNAP gate at a fixed (nontrivial) angle, on a fixed mode s, and acting upon a fixed photon number k, is sufficient to give universality: Theorem 9. Letn, m≥2,0≤k≤n,0≤s≤m−1. Let θ∈Rwithθ̸≡0 mod 2 π. Then the group generated by Un,mandSk,s(θ)is universal. In Corollary 1 of Ref. 11, Bouland proved that if m≥3, one may densely generate Un,m using copies of a single beamsplitter. Combined with Theorem 9, this implies that U(H)may be generated using a single SNAP gate (with fixed mode, angle, and photon number) and a single beamsplitter (acting on different pairs of modes). V . DISCUSSION In Sec. II, we discussed paradigms for the simulation of loss and distinguishability errors in photonics, and we applied these paradigms to investigate the impact of distinguishability errors on Type I fusion, generalized Type II fusion (the n-GHZ state analyzer), and n-GHZ state generation. For low-order errors, one may do the calculations by hand, as in Secs. II D 1 and II D 2. To incorporate higher-order errors in Bell state generation in Sec. II D 3, we turned to numerical simulation. In the longer term, it will be essential to simulate circuits larger than the BSG circuit, an obvious example being n-GHZ state generation. More generally, as in fusion-based quantum computation, one often wants to generate complex linear optical states by starting from small entangled states, such as GHZ
Question 132multiple-choice
Quantum simulation platforms using cold atoms in optical lattices have enabled experimental realization of complex quantum models and advanced algorithmic operations. Programmable control technologies like digital-micromirror devices (DMDs) are expanding the possibilities for engineering quantum Hamiltonians and implementing novel unitary transformations.
Which of the following statements accurately describes a unique advantage of the quadratic quantum Fourier transform (QQFT) protocol on cold atom optical lattice platforms?
1) It allows direct measurement of quantum entanglement without noise.
2) It enables simulation of superconductivity at room temperature.
3) It provides universal quantum computation in a single step.
4) It optimizes laser cooling techniques for higher atom densities.
5) It automatically implements error correction during quantum simulation.
6) It permits programmable engineering of Hamiltonians with complex symmetries and long-range interactions not easily accessible by conventional methods.
7) It eliminates decoherence effects in many-body quantum systems.
✓ Correct Answer:
The correct answer is 6) It permits programmable engineering of Hamiltonians with complex symmetries and long-range interactions not easily accessible by conventional methods..
📚 Reference Text:
Programmable Hamiltonian engineering with quadratic quantum Fourier transform Pei Wang,1,Zhijuan Huang,1,Xingze Qiu,2, 3,and Xiaopeng Li2, 3, 4,y 1Department of Physics, Zhejiang Normal University, Jinhua 321004, China 2State Key Laboratory of Surface Physics, Institute of Nanoelectronics and Quantum Computing, and Department of Physics, Fudan University, Shanghai 200438, China 3Shanghai Qi Zhi Institute, Shanghai 200030, China 4Shanghai Research Center for Quantum Sciences, Shanghai 201315, China (Dated: November 2, 2022) Quantum Fourier transform (QFT) is a widely used building block for quantum algorithms, whose scalable implementation is challenging in experiments. Here, we propose a protocol of quadratic quantum Fourier trans- form (QQFT), considering cold atoms confined in an optical lattice. This QQFT is equivalent to QFT in the single-particle subspace, and produces a di erent unitary operation in the entire Hilbert space. We show this QQFT protocol can be implemented using programmable laser potential with the digital-micromirror-device techniques recently developed in experiments. The QQFT protocol enables programmable Hamiltonian en- gineering, and allows quantum simulations of Hamiltonian models, which are di cult to realize with con- ventional approaches. The flexibility of our approach is demonstrated by performing quantum simulations of one-dimensional Poincar ´e crystal physics and two-dimensional topological flat bands, where the QQFT protocol e ectively generates the required long-range tunnelings despite the locality of the cold atom system. We find the discrete Poincar ´e symmetry and topological properties in the two examples respectively have robustness against a certain degree of noise that is potentially existent in the experimental realization. We expect this approach would open up wide opportunities for optical lattice based programmable quantum simulations. I. INTRODUCTION Quantum Fourier transform1has been widely used in con- structing e cient quantum algorithms, that have exponential quantum speedup over the classical computing2,3. The fa- mous example is Shor’s algorithm, where QFT is integrated in a quantum circuit to perform prime factorization of vi- tal importance to cryptography4. It has been combined with control unitary circuits constituting quantum phase estima- tion5, which can compute many-body Hamiltonian spectra2. It has also been applied to implement quantum phase kick- back in digital quantum simulations of Hamiltonian time evo- lution6. However, its experimental realization meets much challenge7–12, with present noisy-intermediate-scale-quantum devices13. Large scale implementation so-far has not been achieved. Cold atoms confined in optical lattices provide a versa- tile platform for large-scale quantum simulations of quantum many-body physics. There has been tremendous progress in simulating strongly correlated equilibrium physics and exotic quantum dynamics in cold atom experiments. The Fermi- Hubbard model of fundamental importance to modeling cor- related electrons has been implemented with upto hundreds of atoms14–18. Mott-superfluid transition and its quantum criti- cality have been studied in lattices of di erent geometry and of di erent dimensionality19–21. Topological bands and geo- metrical Berry phases in momentum space22–28have been ob- served in artificial gauge field lattices with the e ective mag- netic field strength reaching the order of unit flux quantum per unit cell. The dynamical phase transition from quantum ther- malization to many-body-localization have been investigated with both interaction
Question 133multiple-choice
Quantum algorithms are often employed to solve group-theoretic problems that are computationally challenging for classical computers. Techniques such as oracle simulation, swap tests, and reductions between problems play pivotal roles in designing efficient quantum solutions.
When solving the Hidden Subgroup Problem in the group Zn_p ⋉ Z2 using quantum algorithms, which approach enables a quantum polynomial-time solution by leveraging a reduction to a related problem, and what is the key aspect of this approach?
1) Directly applying the Quantum Fourier Transform to Zn_p ⋉ Z2 without any problem reduction
2) Utilizing Grover's search to find hidden elements in Zn_p ⋉ Z2
3) Reducing the Hidden Subgroup Problem in Zn_p ⋉ Z2 to the Hidden Translation Problem in Zn_p, allowing use of efficient quantum algorithms for translations
4) Mapping Zn_p ⋉ Z2 to a classical group and using classical subgroup enumeration
5) Employing amplitude amplification techniques to increase probability of finding subgroup generators
6) Treating Zn_p ⋉ Z2 as an abelian group and applying abelian HSP algorithms
7) Using quantum walks to identify subgroup structure through state evolution
✓ Correct Answer:
The correct answer is 3) Reducing the Hidden Subgroup Problem in Zn_p ⋉ Z2 to the Hidden Translation Problem in Zn_p, allowing use of efficient quantum algorithms for translations.
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apply Theorem 3.5 in this new setting. Let us now show how to simulate this new oracle access. From |x/angbracketright|b/angbracketright|0/angbracketrightS|0/angbracketrightSwe compute |(−1)bx/angbracketright|b/angbracketright|0/angbracketrightS|0/angbracketrightS, and then we call f0and get|(−1)bx/angbracketright|b/angbracketright|f0((−1)bx)/angbracketrightS|0/angbracketrightS. We multiply the first register by ( −1) and call f1,w h i c hg i v e s |(−1)b+1x/angbracketright|b/angbracketright|f0((−1)bx)/angbracketrightS|f1((−1)b+1x)/angbracketrightS. Finally, we multiply the first register by (−1)b+1a n ds w a pt h el a s tt w or e g i s t e r sw h e n b=1 . As there is a quantum reduction from Hidden Subgroup inZn p/multicloserightZ2toHidden Translation inZn pby the method of [13], we obtain the following corollary. Corollary 3.7. Letpbe a fixed prime. Then Hidden Subgroup (Zn p/multicloserightZ2)can be solved in quantum time poly(n). The algorithm TranslationFinding c a na l s obee x t e n d e dt os o l v e Translating CosetinZn p. Corollary 3.8. Letpbe a prime. Let αbe a group action of Zn p.W h e n t=Ω (p(n+p)p−1log(1/ε)),Translating Coset (Zn p,α,t)c a nb es o l v e di nq u a n t u m time(n+p)O(p)log(1/ε)with error ε. Proof. Let the input of the Translating Coset (Zn p,α,t)b e(|φ0/angbracketright⊗t,|φ1/angbracketright⊗t). We can suppose w.l.o.g. that the stabilizers of |φ0/angbracketrightand|φ1/angbracketrightare trivial. Indeed the stabilizers can be computed by Proposition 2.1. If they are different, then the algo- rithm obviously has to reject; otherwise we work in the factor group Zn p/G|φ0/angbracketright∼=Zn/prime p for some n/prime≤n. To be more specific, we can compute a ( Zp-basis for) a subgroup G1ofZn pwhich is a direct complement of G|φ0/angbracketrightby augmenting a basis for G|φ0/angbracketrightto a basis ofZn p, and we can actually work with G1in place of G. Forb=0,1, letfbbe the injective quantum function on Gdefined by |fb(x)/angbracketright= |x·φb/angbracketrightfor every x∈G. If the translating coset of ( |φ0/angbracketright,|φ1/angbracketright)i se m p t y ,t h e n f0and HIDDEN TRANSLATION AND TRANSLATING COSET 13 f1have distinct ranges. Otherwise the translating coset of ( |φ0/angbracketright,|φ1/angbracketright) is a singleton {u},a n d(f0,f1) have the translation u. Thealgorithmfor Translating Coset oninput ( |φ0/angbracketright⊗t,|φ1/angbracketright⊗t)isthealgorithm TranslationFinding on input ( f0,f1) with a few modifications described below. The oracle access to ( f0,f1) is modified in the same way as in Corollary 3.6. We simulate theith query |x/angbracketright|b/angbracketright|0/angbracketrightS|0/angbracketrightSusing the ith copy of |φ0/angbracketright|φ1/angbracketright.T h et w or e g i s t e r s |0/angbracketrightS|0/angbracketrightS are swapped with |φb/angbracketright|φ1−b/angbracketright, and then we let act xon|φb/angbracketrightand (−x)o n|φ1−b/angbracketright. The equality tests in steps 0 and 9 are rep laced by the swap test [7, 18] iterated O(log(1/ε)) times. Finally, Nis multiplied by O(log(1 /ε)), and the algorithm re- jects whenever the algorithm TranslationFinding aborts or there is no solution in step 9. 4. Translating coset in solvable groups. 4.1. Preparation. 4.1.1. Quantization of the problems. LetGbe a black-box group with unique encoding, and let αbe a group action on Γ. We now describe quantum analogues of problems with classical outcomes, as unitary transformations whose outputs are basically uniform superpositions on the possible classical outcomes. We
Question 134multiple-choice
Hybrid tensor networks are an emerging approach in quantum simulation, combining quantum resources with classical computational techniques to efficiently model complex quantum systems. These methods are particularly relevant for simulating large many-body wave functions with limited quantum hardware.
Which key advantage does a hybrid tensor network offer for simulating large quantum systems on current intermediate-scale quantum computers?
1) Requires only classical resources for all calculations
2) Eliminates the need for tensor contractions entirely
3) Allows simulation of only small quantum systems due to device limitations
4) Restricts applications to quantum field theory exclusively
5) Guarantees fault-tolerance on noisy quantum devices
6) Uses only quantum hardware with no classical computation
7) Enables simulation of systems larger than the available quantum hardware by leveraging classical tensors alongside quantum states
✓ Correct Answer:
The correct answer is 7) Enables simulation of systems larger than the available quantum hardware by leveraging classical tensors alongside quantum states.
📚 Reference Text:
Title: Quantum Simulation with Hybrid Tensor Networks. Year: 2020 Paper ID: d7f557f1e1edb1da11950a26cca6137398352c10 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/d7f557f1e1edb1da11950a26cca6137398352c10 Abstract: Tensor network theory and quantum simulation are, respectively, the key classical and quantum computing methods in understanding quantum many-body physics. Here, we introduce the framework of hybrid tensor networks with building blocks consisting of measurable quantum states and classically contractable tensors, inheriting both their distinct features in efficient representation of many-body wave functions. With the example of hybrid tree tensor networks, we demonstrate efficient quantum simulation using a quantum computer whose size is significantly smaller than the one of the target system. We numerically benchmark our method for finding the ground state of 1D and 2D spin systems of up to 8×8 and 9×8 qubits with operations only acting on 8+1 and 9+1 qubits, respectively. Our approach sheds light on simulation of large practical problems with intermediate-scale quantum computers, with potential applications in chemistry, quantum many-body physics, quantum field theory, and quantum gravity thought experiments.
Question 135multiple-choice
Quantum algorithms offer new approaches to signal and image processing by leveraging the principles of superposition and parallelism, potentially enabling efficient manipulation of large-scale visual data. In classical methods, techniques like JPEG compression exploit localized correlations within images by processing blocks separately.
Which feature distinguishes the JPEG-inspired quantum interpolation algorithm from classical bicubic interpolation, particularly in terms of computational complexity and parallelism?
1) It requires exponentially increasing circuit depth as image size grows.
2) It applies transformations sequentially to each pixel of the image.
3) Its performance metrics are consistently superior to classical bicubic interpolation for all images.
4) It does not exploit any blockwise correlations within images.
5) It relies on quantum measurement to maintain parallel processing advantage.
6) It achieves constant circuit depth regardless of image size by applying parallel transformations to all blocks encoded in the least significant qubits.
7) It compresses images using wavelet transforms instead of discrete cosine transforms.
✓ Correct Answer:
The correct answer is 6) It achieves constant circuit depth regardless of image size by applying parallel transformations to all blocks encoded in the least significant qubits..
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QCT can be applied to all members of the superposition insingularcost. However, weneedtokeepinmindthatif the data in superposition needs to then be extracted via measurements, theadvantagethattheparallelprocessing power introduces will be lost. IV. SUBSPACE-QCT RESAMPLING OF NATURAL IMAGES The power of parallel QFT computation can be ex- tended beyond actions on different sets of data encoded in superposition. Subsets of a single data entry can also be processed in parallel. Correlations present in natural data tend to be local- ized within short distances. Indeed, natural images are usually built from large structures (in number of pix- els) that do not necessarily correlate with the rest of the picture. The JPEG protocol for image compression [4] exploits this fact by performing DCT onto 88blocks of the image. This way, the small DCT only capture short range correlations, and the following steps of the protocol can be described using 88matrices. A similar procedure can be applied in a quantum cir- cuit. All the 88pixel matrices of the image encoded in the proposed way are stored in the superposition of the 3 least significant qubits. The rest of the system can be un- derstood in the same way we considered the label register for multi-layer images. Applying a transformation to the 3 least significant qubits, a QCT in this case, achieves in constant depth the transformation of all subspaces of the image. To be precise, this JPEG-inspired interpolation algo- rithm using the DCT proceeds as follows. Given an im- age encoded into a quantum state in registers jxijyijli, perform the QCT on the last 3 qubits of registers jxi, and simultaneously in the last 3 qubits of register jyi. Introduce the ancilla qubits needed for the interpolationm=1Bicubic QFT n-QCT 3-QCT PSNR30.095 27.395 29.930 29.988 SSIM0.880 0.829 0.871 0.878 Table I. Comparison of the Peak Signal-to-Noise Ratio (PSNR) and Structural SIMilarity (SSIM) of the gray-scale cameraimage, after different methods of interpolation. The image is downsized to half its original size ( m= 1) via a classical algorithm using the pixel area relation of the image implemented in OpenCV. The image is interpolated to the original size and compared to the base image. The classical bicubic inteprolation scheme yield the best PSNR and SSIM values, closely followed by the s=3-QCT. From the quantum algorithms, the best choice is the JPEG-inspired s=3-QCT method, it provides the best results and is the most efficient option. after the third least significant qubit on both registers. Undo the QCT transformation on the extended space. This achieves image interpolation at a depth constant with the original system size. Given a fixed subspace 2s, s= 3in the JPEG-inspired procedure, this algorithm resamples an 2nsignal into a 2n+mspace in complexity O (s+m)2 , the complexity of the algorithm no longer depends on the size of the original image. A circuit de- picting this algorithm is sketched in Fig. 4b. This algorithm can be generalized to any subspace s, and can introduce improvements depending on the un- derlying structure of the signal. In particular, an
Question 136multiple-choice
In quantum computing and group theory, the hidden subgroup problem (HSP) and its variants play a pivotal role in algorithm design and complexity theory. The structure and representations of finite groups are central to understanding these algorithms.
Which mathematical property ensures that the regular representation of a finite group decomposes into a direct sum of all its irreducible representations, each appearing with multiplicity equal to its dimension?
1) The regular representation is reducible and decomposes according to the group's representation theory.
2) The group is necessarily Abelian, so all representations are one-dimensional.
3) The regular representation is always irreducible for non-Abelian groups.
4) Only groups with normal subgroups have decomposable regular representations.
5) The decomposition requires the group to be cyclic.
6) The decomposition occurs only for quantum groups, not classical finite groups.
7) The regular representation decomposes only if the group is infinite.
✓ Correct Answer:
The correct answer is 1) The regular representation is reducible and decomposes according to the group's representation theory..
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the hidden subgroup conjugacy problem, but less is known about the reverse relation. F irst it is clear that when the group is Abelian or when the subgroups are normal, the HSP is equ ivalent to the HSCP (since subgroups of Abelian groups and normal subgroups are con jugate only to themselves.) Recently Fenner and Zhang[45] have examined the difference betwe en the search and decision version of the HSP (in the search problem one is required to return t he hidden subgroup and in the decision subgroup one is required to distinguish whether th e hidden subgroup is trivial or not.) Their results imply that the HSCP and the HSP over per mutation groups are polynomial time equivalent. Similarly they show that for the dihedr al group, when the order of the group is the product of many small primes, then the HS CP and the HSP are polynomial time equivalent. In Section 6 we will show that the HSCP and HSP are quantum polynomial time equivalent for the Heisenberg group. Finally we note that the HSP can be decomposed into the HSCP along wit h what we call the hidden conjugate subgroup problem (HCSP). In the hidden con jugate subgroup, one is given a function which hides one of a set of subgroups all of which are conjugate to each other and one desires to identify the conjugate subgroup. For the single copy HCSP, the pretty good measurement was shown to be optimal by Moore and Russell[22]. 4 Symmetry Considerations and the HSP The hidden subgroup state of Eq. (2) possess a set of symmetries which allow us to, with- out loss of generality, perform a change of basis which exploits thes e symmetries. These symmetries are related to the regular representations of the gro up. 4.1 Regular Representations There are two regular representations of the group Gwhich we will be interested in, the left regular representation and the right regular representation. Bo th of these representations act on a Hilbert space with a basis labeled by the elements of the group G. Define the left regular representation via its action on basis states of this Hilbert space, RL(g)|g′/an}bracketri}ht=|gg′/an}bracketri}ht, (5) wheregg′is the element of Gobtained by multiplying gandg′. Similarly, define the right regular representation via RR(g)|g′/an}bracketri}ht=|g′g−1/an}bracketri}ht. (6) The regularrepresentationsare, in general, reducible represent ationsof G. In fact these repre- sentations are particularlyimportant in the representationtheor y offinite groups. The reason for this is that the regular representations are reducible into a dire ct sum of all irreducible representations (irreps) of the group G. Thus, it is possible to find a basis in which RLacts as RL(g) =/circleplusdisplay µIdµ⊗Dµ(g), (7) D. Bacon 9 where the direct sum is over all irreps of the group G,Dµ(g) is theµth irrep evaluated at group element ganddµis the dimension of the irrep µ. A similar decomposition occurs for the right regular representation, RR(g) =/circleplusdisplay µDµ(g)⊗Idµ. (8) In fact, an elementary result of finite group representation theo ry tells us that the basis in
Question 137multiple-choice
In the representation theory of unitary and symmetric groups, combinatorial objects such as Standard and Semi-Standard Young Tableaux (SYT and SSYT) play a crucial role in labeling basis vectors and understanding group actions. These tableaux have specific rules governing their entries and structure, which are fundamental to their application in mathematical physics and combinatorics.
Which statement correctly describes a necessary property of Semi-Standard Young Tableaux (SSYT) with entries from {1, 2,.., d} as used to label basis vectors of irreducible representations of the unitary group U_d?
1) Entries must increase strictly across rows and columns.
2) No two boxes with different labels can appear in the same row.
3) Each label must appear exactly once in the tableau.
4) Columns are weakly increasing while rows are strictly increasing.
5) Rows are weakly increasing and columns are strictly increasing.
6) All boxes labeled with the same integer must be in the same row.
7) The number of boxes in each row must equal the number of boxes in each column.
✓ Correct Answer:
The correct answer is 5) Rows are weakly increasing and columns are strictly increasing..
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1interchanged, which can be seen to be a SYT. If kandk+ 1are in the same row, they must be next to each other and the action is given as (k,k+ 1)|T/angbracketright=|T/angbracketright, (2.7) and if they are in the same column (and necessarily in adjacent rows) the action is (k,k+ 1)|T/angbracketright=−|T/angbracketright. (2.8) For the unitary group, a subgroup tower that leads to a canonical basis is 1 =U1⊂U2⊂...Ud, whereUi is the unitary group acting on the i×iminor of the full d×dmatrix. This tower is determined once we x a basis for each qudit register. This tower, like the one for the symmetric group, gives rise to multiplicity-free branching and hence to a canonical basis. This canonical basis is called the Gelfand-Tsetlin basis. Given any irrepλofUdin the form of a Young diagram, one can obtain the diagrams in the restriction to Ud−1 by removing a box from end of each column in all possible ways. If two columns have the same length in λ and the choice is to remove a box from the left column, then one must also remove the box from the right column to ensure that a valid Young diagram is obtained. An example is given below. λ=→ ,,, . (2.9) 5 Suppose we pick one and proceed with a choice down the tower, we would have the following possibility. →→→. (2.10) This sequence gives us a basis vector of the irrep λ. This can be encapsulated by putting numbers into the original irrep, which represent the stage before which the boxes are removed. For the above sequence the following numbering would hold. 112 233 3. (2.11) Notice that the rows are weakly increasing i.e., the numbers either increase or stay the same as we move right and the columns are strictly increasing. Such a Young diagram is called a semi-standard Young diagram (SSYT). SSYTs with numbers taken from the set [d]label basis vectors in the irrep of Ud. We will see below that SSYTs also play a role in certain induced representations of the symmetric group called permutation modules. Ecient encodings of these bases are constructed in [5] i.e., using poly( logd, n,log 1//epsilon1) bits. For a SSYT, the tuple that contains the number of boxes labeled by a given integer is called the content of the SSYT. For instance, in the example above, the content is (2,2,3)corresponding to 2boxes numbered one, 2boxes numbered two and 3boxes numbered three. SSYTs have an interesting structure that is useful in our algorithms. If we consider all the boxes con- taining a speci c number, we nd that no two of them appear in the same column. If we isolate these boxes, such a skew diagram is called a horizontal strip . An SSYT can be thought of as being composed of horizontal strips. It turns out that this composition can be made more precise as we describe in the next subsection. An example of an SSYT and the associated horizontal strips are given below. 112 233 3,11,2
Question 138multiple-choice
In the representation theory of the symmetric group Sn, central elements in the group algebra and their action on irreducible representations play a crucial role in analyzing operators such as permutation-invariant Hamiltonians. The behavior of these operators can be described using character theory and combinatorial formulas.
For the irreducible representation of Sn labeled by the partition λ = [n−k, k], what is the explicit formula for the scalar ηλ by which the sum over all transpositions acts on this irrep?
1) ηλ = k(n−k)
2) ηλ = n(n−1)/2
3) ηλ = k(n+1) − k^2
4) ηλ = 2k(n+1) − 2k^2
5) ηλ = n(n+1) − 2k^2
6) ηλ = 2k(n−1) − 2k^2
7) ηλ = k^2(n+1) − k
✓ Correct Answer:
The correct answer is 4) ηλ = 2k(n+1) − 2k^2.
📚 Reference Text:
(2.32) whereηλis some scalar depending only on the irrep λandIis an identity matrix of the appropriate dimension. Proof. We begin by showing something stronger: let qKn=/summationdisplay (i,j)∈E(Kn)(i j)∈C[Sn] (2.33) denote the sum over all transpositions in Sn. Then, for any permutation π∈Sn, we have πqKnπ−1=π/summationdisplay i,j=1,...,n(i j)π−1=/summationdisplay i,j=1,...,n(π(i)π(j)) =qKn (2.34) from which it follows that the element qKnis central in the group algebra C[Sn]. In particular, for any irrep λ= [n−k,k], we see that ρλ(qKn) is central in the irreducible algebra of matrices ρλ(C[Sn]) acting on the vector space Vλas defined in Eq. (2.8). Similarly for any such ρλwe seeρλ(2/parenleftbign 2/parenrightbige−2qKn) =Hλ Knis anSn-linear map from the irreducible module Vλto itself and hence, by Schur’s lemma, is a scalar multiple of the identity matrix. This completes the proof. With a little bit of extra work, we can also compute the scalar multiple ηλfrom Eq. (2.32) associated with Hamiltonian of the clique in each irrep. We do this next. Lemma 2.12. Letηλbe as in Lemma 2.11. Then, for any integer nand irrepρ[n−k,k]we have η[n−k,k]= 2k(n+ 1)−2k2(2.35) Proof. We can find the value of ηλby computing two quantities: the dimension of the λirrep (i.e., the trace of ρλ(e)) and the trace of Hλ Kn. It is possible to compute both of these quantities with the Frobenius trace formula, but we shall save some effort by instead Accepted in Quantum2024-03-25, click title to verify. Published under CC-BY 4.0. 16 using the hook length formula and Murnaghan-Nakayama rule. To deal with the many partial factorials appearing in this calculation, recall the falling factorial notation n[k]from Eq. (2.29). We shall also use the dimension of the [ n−k,k] irrep given in Lemma 2.10. For ease of future use we also write this formula as χ[s,t](e) =s−t+ 1 s+ 1/parenleftigg s+t t/parenrightigg =(s−t+ 1)(s+t)[t−1] t!. (2.36) Next we compute the trace of Hλ Kn. We can do this by computing the character χλ((i j)) of a single transposition ( i j) in theλirrep. Since all transpositions belong to the same conjugacy class, they will also have the same character. We now compute this character. The Murnaghan-Nakayama rule relates the character of a permutation in some irrep to the character of smaller permutations in irreps of the symmetric group on fewer elements. In the special case considered here, it gives χλ((i j)) =/summationdisplay ξ(−1)h(ξ)χλ/ξ(e) (2.37) where the sum runs over all ways of removing two adjacent boxes from the irrep λwhile leaving a valid Young diagram; h(ξ) is one if the boxes removed are stacked vertically, zero otherwise; and λ/ξis the resulting partition when the boxes are removed from λ. More explicitly, we write λ= [n−k,k] and then, for any n>3: χ[n−k,k]((i j)) = χ[n−k−2,k](e) if k= 1 χ[n−k−2,k](e) +χ[n−k,k−2](e) if 2≤k≤n/2−1 χ[n−k,k−2](e) if k= (n−1)/2 χ[n−k,k−2](e)−χ[n−k−1,k−1](e) ifk=n/2.(2.38) By Eq. (2.14) the Hamiltonian for a clique Knis HKn= 2/summationdisplay (i,j)∈E(Kn)I−2/summationdisplay (i,j)∈E(Kn)Swapij (2.39) = 2/parenleftigg n 2/parenrightigg I−2/summationdisplay (i,j)∈E(Kn)ρ[n−k,k]((i j)), (2.40) and taking the trace of this expression gives Tr/bracketleftig H[n−k,k] Kn/bracketrightig = 2/parenleftigg n 2/parenrightigg χ[n−k,k](e)−2/summationdisplay
Question 139multiple-choice
Algebraic geometry and quantum field theory intersect in the modeling of topological quantum computation, where the structure of symmetries and algebraic properties determines the behavior of quantum systems. The construction of Gröbner bases and the representation of the Braid group are key for understanding anyon statistics and the scalability of quantum algorithms.
Which modification allows the reconstruction of a faithful Braid group representation in quantum computing models with indefinite metrics, after initial limitations due to twisted inner automorphisms and parity invariance?
1) Introducing time-reversal symmetry before associativity in the axiomatic framework
2) Applying quantum deformations to toric varieties along with a specific coordinate prescription
3) Utilizing classical coordinate systems on projective varieties without deformation
4) Replacing the cluster property with locality in the axiomatic sequence
5) Constructing Gröbner bases solely with lexicographic monomial ordering
6) Removing parity invariance from the Wightman axioms
7) Defining the Braid group via ideal quotients in order domains
✓ Correct Answer:
The correct answer is 2) Applying quantum deformations to toric varieties along with a specific coordinate prescription.
📚 Reference Text:
Title: Gröbner bases for finite-temperature quantum computing and their complexity Year: 2010 Paper ID: 928556d94291a6d5ec167a5c4650514d92bee34b Source: semantic-scholar URL: https://www.semanticscholar.org/paper/928556d94291a6d5ec167a5c4650514d92bee34b Abstract: Following the recent approach of using order domains to construct Grobner bases from general projective varieties, we examine the parity and time-reversal arguments relating to the Wightman axioms of quantum field theory and propose that the definition of associativity in these axioms should be introduced a posteriori to the cluster property in order to generalize the anyon conjecture for quantum computing to indefinite metrics. We then show that this modification, which we define via ideal quotients, does not admit a faithful representation of the Braid group, because the generalized twisted inner automorphisms that we use to reintroduce associativity are only parity invariant for the prime spectra of the exterior algebra. We then use a coordinate prescription for the quantum deformations of toric varieties to show how a faithful representation of the Braid group can be reconstructed and argue that for a degree reverse lexicographic (monomial) ordered Grobner basis, the complexity class of this problem i...
Question 140multiple-choice
In computational quantum chemistry and materials science, efficiently generating localized orbitals from large electronic structure calculations is critical for interpreting chemical bonding and reducing computational cost. Advanced algorithms often incorporate mathematical properties and physical intuition to optimize performance in these tasks.
Which strategy most directly accelerates repeated orbital localization in large systems by enabling parallelism and focusing computation on regions of significant electron density?
1) Performing exhaustive global QR factorizations on the entire density matrix
2) Partitioning the column selection into local QRCP factorizations guided by electron density and using randomized sampling
3) Increasing the frequency of I/O operations to improve data throughput
4) Ignoring electron density and sampling columns uniformly across all regions
5) Selecting columns based solely on their numerical magnitude without regard to conditioning
6) Using only the original orthonormal basis for all calculations
7) Restricting computations to vacuum regions where electron density is minimal
✓ Correct Answer:
The correct answer is 2) Partitioning the column selection into local QRCP factorizations guided by electron density and using randomized sampling.
📚 Reference Text:
regardless of the version used. Another key feature of the algorithm, especially for our modi cations later, is that because we are e ectively working with the spectral projector P;the method performs equivalently if a di erent orthonormal basis for the range of is used as input. In physics terminology, the SCDM procedure is gauge-invariant. Lastly, the key factor in forming a localized basis is the selection of a well conditioned subset of columns. Small changes to the selected columns, provided they remain nearly as well conditioned, may not signi cantly impact the overall quality of the basis. 3. The approximate column selection algorithm. When the SCDM proce- dure is used as a post-processing tool for a single atomic con guration, the computa- tional cost is usually a ordable. In fact in such a situation, the most time consuming part of the computation is often the I/O related to the matrices especially for systems of large sizes. However, when localized orbitals need to be calculated repeat- edly inside an electronic structure software package, such as in the context of hybrid functional calculations with geometry optimization or ab initio molecular dynamics simulations, the computational cost of SCDM can become relatively large. Here we present an algorithm that signi cantly accelerates the SCDM procedure. The core aspect of the SCDM procedure is the column selection procedure. Given a set of appropriate columns the requisite orthogonal transform to construct the 6 SCDM may be computed from the corresponding rows of , as seen in Algorithm 2. Here we present a two stage procedure for accelerating this selection of columns and hence the computation of . First, we construct a set of approximately localized orbitals that span the range of via a randomized method that requires only and the electron density ;though ifis not given it may be computed directly from without increasing the asymptotic computational complexity. We then use this approximately localized basis as the input for a procedure that re nes the selection of columns from which the localized basis is ultimately constructed. This is done by using the approximate locality to carefully partition the column selection process into a number of small, local, QRCP factorizations. Each small QRCP may be done in parallel, and operates on matrices of much smaller dimension than : 3.1. Approximate localization. The original SCDM procedure, through the QRCP, examines all Ncolumns of to decide which columns to use to construct Q. However, physical intuition suggests that it is often not necessary to visit all columns to nd good pivots. For instance, for a molecular system in vacuum, it is highly unlikely that a pivot comes from a column of the density matrix corresponding to the vacuum space away from the molecule. This inspires us to accelerate the column selection procedure by restricting the candidate columns. This is accomplished by generating O(nelogne) independent and identically dis- tributed (i.i.d.) random sample columns, using the normalized electron density as the probability distribution function (pdf). Indeed, if a column of
Question 141multiple-choice
In computational group theory, the Hidden Subgroup Problem (HSP) and its generalization, the Hidden Symmetry Subgroup Problem (HSSP), play key roles, especially in the context of group actions and algorithmic reductions. Frobenius groups, with their distinct structure and permutation properties, are of particular interest for efficiently solving such problems.
Which property of Frobenius groups enables a polynomial-time reduction from HSSP to HSP, thereby facilitating the efficient solution of certain computational problems like the Hidden Quadratic Polynomial Problem?
1) The kernel subgroup K is always abelian in Frobenius groups.
2) Every subgroup of a Frobenius group is cyclic.
3) Frobenius groups are always simple groups with no normal subgroups.
4) The action of the group on the kernel K is always regular.
5) The complement subgroup H is always unique up to isomorphism.
6) All elements outside the kernel K have order two.
7) The existence of an efficiently computable H-strong base in the set M.
✓ Correct Answer:
The correct answer is 7) The existence of an efficiently computable H-strong base in the set M..
📚 Reference Text:
finite group, Ma finite set, ◦a polynomial time computable action of GonM,a n dHa family of subgroups of G. If there exists an efficiently computable H-strong base in M,t h e n HSSP(G,M,◦,H)is polynomial time reducible to HSP(G,H). Together with Proposition 3.1, this result demonstrates that, in contrast to or- dinary bases, finding (and even understanding the existence of) strong bases can be quite difficult. The results in the rest of the paper rely on constructing strong basesin two different contexts. 4. The HSSP for Frobenius complements and the HQPP. In view of Proposition 3.4, we are interested in group actions for which there exist easily com-putable (and therefore also small) bases for some interesting families of subgroups. If in addition the related HSPis easy to solve, then we have efficiently solvable HSSPs. It turns out not only that Frobenius groups under some conditions have these prop- erties, but also that the HQPPcan be cast as one of these HSSPs. 4.1. Strong bases in Frobenius groups. AFrobenius group is a transitive permutation group acting on a finite set such that only the identity element has more than one fixed point and some nontrivial element fixes a point (see, for example, [23]). Let us recall here some notions and facts about these groups. Let Gbe a Frobenius group with action ◦ MonM. The identity element together with the elements of G that have no fixed points form a normal subgroup K,t h eFrobenius kernel , for which we also have |K|=|M|. This latter fact and that Kis closed under conjugation are easy to prove. Surprisingly, all the known proofs for the statement that Kis a subgrouprequirerepresentationtheory. Asubgroup HofGisaFrobenius complement if it is the stabilizer Hmof some element m∈M. It is a subgroup complementary toK,t h a ti s , K∩H={1}andG=KH. Hence, the group Gis a semidirect product K/multicloserightHofKandH.W ed e fi n et h eb i n a r yo p e r a t i o n ◦K:G×K→Kby g◦Kx=yhxh−1, whenx∈Kandg=yhwithy∈Kandh∈H. Itisastraightforwardcomputationto check that ◦Kis an action of GonK.O b s e r v et h a t Kacts on itself by multiplication from the left, while Hacts onKby conjugation. Furthermore, we can identify the action◦Mwith the action ◦Kvia the map φ:M→Kdefined as follows. For any n∈M,t h e r ee x i s t s gn∈Gsuch that gn◦Mm=nsinceGis transitive. If gn=ynhn withyn∈Kandhn∈H, by definition we set φ(n)=yn.N o t et h a t φdepends on the choice of m, or, equivalently, on the choice of the complement H.O b s e r v ea l s o thatφ(n) can be characterized as the unique element ynofKwithyn◦Mm=n, and therefore φis a bijection. Then indeed for every g∈Gandn∈M,w eh a v e g◦Kφ(n)=φ(g◦Mn). From now on we will suppose without loss of generality that the action is ◦K, which we denote for simplicity by ◦. HIDDEN SYMMETRY SUBGROUP PROBLEMS 1995 Observethen that with respectto ◦, the Frobenius complement His the stabilizer ofe, the identity element of K. The orbits of Hare{e}and some
Question 142multiple-choice
Isogeny-based cryptography relies on the hardness of finding isogenies between supersingular elliptic curves, but recent advances have revealed vulnerabilities in widely used protocols. Cryptographic schemes in this area often balance security and efficiency by carefully choosing which mathematical data to reveal during key exchange.
Which cryptanalytic breakthrough demonstrated that SIDH protocols with balanced parameters and known endomorphism ring can be broken in polynomial time using superspecial abelian surfaces?
1) The development of group actions on supersingular elliptic curves (KMPW21)
2) The introduction of suborder representations for large prime degree isogenies (pSIDH)
3) The proposal of an isogeny representation based on torsion point images
4) The SCALLOP scheme using partial isogeny representations
5) The computation of the endomorphism ring of the codomain to break pSIDH
6) Robert's attack on SIDH with unknown endomorphism ring
7) The Castryck-Decru and Maino-Martindale polynomial-time attacks using superspecial abelian surfaces
✓ Correct Answer:
The correct answer is 7) The Castryck-Decru and Maino-Martindale polynomial-time attacks using superspecial abelian surfaces.
📚 Reference Text:
alreadyincluded parametersetswhichcouldhave been used in SIDH variants.Neverthelessnone ofthese attacksd irectly impacted SIDH where AandBare roughly the same size. Then in 2022 Castryck and De- cru [CD22] (and independently Maino and Martindale [ MM22]) vastly improved these using ingenious techniques (utilizing superspecial abelian surf aces) which break SIDH with known endomorphism ring in polynomial time even if AandB are balanced. Finally, Robert proposed a polynomial-time attack on S IDH with unknown endomorphism ring (furthermore, he only needs B2> Aas opposed toB >Ain other attacks). These attacks have shown that using smooth degree isogenies and providing torsion point information will potentially not lead to secure and efficien t cryp- tographic constructions (in [ FMP23] some countermeasures are proposed, but the ones that are not broken are much less efficient than the origina l SIDH con- struction). Thus, in order to navigate in the supersingular isogeny graph parties have to share some other kind of extra information. Alternative isogeny representations. In the pSIDH protocol introduced by Leroux [Ler22a], one reveals suborder representations for isogenies of large prime degreestobuildakeyexchange.Suborderrepresentationsarea particularkindof weak isogeny representations, i.e. some data to represent isogen ies together with an algorithm to efficiently evaluate these isogenies on any point up to a scalar. Prime degree isogenies were not really used before as one cannot wr ite down the isogeny itself (but one can compute its codomain with non-trivial tec hniques). More recently, a similartype ofsecret isogenywas used in the SCALL OPscheme [DFFK+23]. In SCALLOP, a partial isogeny representation is revealed to the attacker. From a cryptanalytic point of view, the unlimited amount of torsion inf or- mation provided by the isogeny representation revealed in pSIDH (a nd more generally, any isogeny representation) is very interesting. Howev er, when the kernel points are not defined over a small extension, the known alg orithms do not apply and it is still unclear how to exploit the isogeny representat ion to recover the secret isogeny. 4 M. Chen, M. Imran, G.Ivanyos, P.Kutas, A.Leroux, C. Petit Lerouxstudied the case where a specific isogenyrepresentation( the suborder representation) is revealed, but we can generalize this setting to a ny isogeny rep- resentation. He showed that computing the endomorphism ring of t he codomain wouldmakepSIDH insecure, thereforemotivating Problem1.1in the p rime case. Morerecently,Robertintroducedyetanotherisogenyrepresen tationbasedon torsion point images and the recent SIDH attacks [ Rob22]. This representation could be used (for isogenies with large prime degrees) instead of the suborder representation to derive a key exchange protocol similar to pSIDH , and this protocol would be similarly affected by our new results. A group action for SIDH and pSIDH In [KMPW21 ] the authors introduce a group action on a particular set of supersingular elliptic curves. Le tEbe a supersingular elliptic curve with endomorphism ring isomorphic to O. Then (O/NO)∗acts naturallyon the set of cyclicsubgroupsof EoforderN. Ifthere is a one-to-one correspondence between cyclic subgroups and N-isogenous curves, then one can look at this action as acting on a set of curves. This act ion was used to provide a
Question 143multiple-choice
In the study of finite abelian groups and their automorphisms, matrix representations are a powerful computational tool, especially when analyzing group characters and transformations. These concepts are essential in quantum algorithms that leverage group structure for problem-solving efficiency.
When representing automorphisms of finite abelian p-groups using matrices, which statement correctly describes the relationship established between character evaluations and matrix conjugation?
1) Character evaluations remain unchanged under any non-singular matrix transformation of group elements.
2) Character evaluations are always inversely related when automorphisms are represented by diagonal matrices.
3) Evaluating a character on a transformed group element always results in a zero value unless the automorphism is the identity.
4) The equivalence of character evaluations relies solely on the group being cyclic.
5) Evaluating the character χx on Φ(d) is equivalent to evaluating the conjugate character χˆΦ(x) on d via matrix conjugation.
6) The use of matrix representations eliminates the need for character theory in quantum algorithms.
7) Direct products of cyclic groups cannot be represented with matrix transformations for automorphisms.
✓ Correct Answer:
The correct answer is 5) Evaluating the character χx on Φ(d) is equivalent to evaluating the conjugate character χˆΦ(x) on d via matrix conjugation..
📚 Reference Text:
M. R¨ otteler and T. Beth. Polynomial-time solution to t he hidden subgroup problem for a class of non-abelian groups. arXiv:quant-ph/9812070. [29] K. Shoda. ¨UberdieAutomorphismen einer endlichen Abelschen Gruppe. Math. Ann. , 100:674– 686, 1928. [30] P. W. Shor. Algorithms for quantum computation: Discre te log and factoring. In Proc. 35th Annual IEEE Symposium on Foundations of Computer Science , pages 124–134, 1994. [31] J. Watrous. Quantum algorithms for solvable groups. In Proc. 33rd Annual ACM Symposium on Theory of Computing , pages 60–67, 2001. arXiv:quant-ph/0011023. [32] H. P. Yuen, R. S. Kennedy, and M. Lax. Optimum testing of m ultiple hypotheses in quantum detection theory. IEEE Trans. Inform. Theory , 21:125–134, 1975. 16 A Matrix representation of Φ(b) In this appendix, we show that Φ(b)andˆΦ(b)can be represented by (simply related) matrices, and furthermore, that χx(Φ(b)(d)) =χˆΦ(b)(x)(d). Lemma 7. LetAbe a finite abelian group, ϕ∈Aut(A),b∈Nwith the corresponding Φ :A→A defined by Φ :=/summationtextb−1 i=0ϕi, andχ:A→Ca character of A. Then there exists a function ˆΦ :A→A such thatχx(Φ(d)) =χˆΦ(x)(d)for alld,x∈A. Proof.LetA∼=Ap1×···×Aprbetheelementary divisor decomposition of A, i.e., thedecomposition intop-groups where the piare distinct primes. Accordingly, let d= (d1,...,dr),x= (x1,...,x r), andχ(i) xi:Api→Csuch thatχx(d) =χ(1) x1(d1)···χ(r) xr(dr). It is known that any automorphism ϕ∈Aut(A) can be decomposed as ϕ= (ϕ1,...,ϕ r) withϕi∈Aut(Api) [29]. Similarly, we have the decomposition Φ( d) = (Φ 1(d1),...,Φr(dr)). Hence if we prove the lemma for p-groups, then this proves it for all finite abelian groups. Assume therefore that Ais ap-group,A∼=Zpe1×··· ×Zpekwithe1≥ ··· ≥ek. By [29], we can represent ϕ∈Aut(A) as a matrix transformation ( x1,...,x k)/ma√st≀→(x1,...,x k)µwhereµ∈Zk×k andpej−ei|µijfor alli>j. Consequently, we can also represent the transformation Φ b y a matrix Mwithpej−ei|Mijfor alli>j. Now define a conjugate matrix ˆMji:=pei−ejMijfor alli,j(note that all entries of ˆMare integers). For any character χx:A→C, we find χx(Φ(d)) = exp/parenleftig 2πi/summationdisplay jidiMijxj/pej/parenrightig = exp/parenleftig 2πi/summationdisplay ijdixjˆMji/pei/parenrightig =χˆΦ(x)(d),(49) whereˆΦ :A→Ais the matrix transformation defined by ( x1,...,x k)/ma√st≀→(x1,...,x k)ˆM. Note that if A=ZN(as in Section 5), then Φ can be represented by a single scalar M∈ZN, and hence ˆΦ = Φ. Also, if A=Zr p(as in Section 6), then M∈Zr×r pandˆM=MT. For notational simplicity, since we only need the matrix representation of ˆΦ (and not of Φ) in these two cases, we putµ→µTandM→MTthroughout the body of the paper. B Proof of Lemma 5 Proof.Forx∈(Zn p)k,w∈Zn p, andb∈Zk p, let Ξx w(b) = 0 denote the system of polynomial equations (48). We want to understand the typical behavior of ηx w:=|{b: Ξx w(b) = 0}|=/summationdisplay bδ[Ξx w(b) = 0] (50) for uniformly random x,w. Specifically, we want to show that it is typically close to it s mean, µ:=E x∈Ak,w∈A[ηx w] =pk−r(51) where we have used (14). To compute the variance, note that E x∈Ak,w∈A[(ηx w)2] =1 prk+r/summationdisplay x,w(ηx w)2(52) =1 prk+r/summationdisplay x,w/parenleftbigg/summationdisplay bδ[Ξx w(b) = 0]/parenrightbigg/parenleftbigg/summationdisplay cδ[Ξx w(c) = 0]/parenrightbigg (53) =1 prk+r/summationdisplay x,w/parenleftbigg/summationdisplay bδ[Ξx w(b) = 0]+/summationdisplay b/negationslash=cδ[Ξx w(b) = Ξx w(c) = 0]/parenrightbigg .(54) 17 The first (diagonal) term is just the mean. For the second (off-d iagonal) term, note that the
Question 144multiple-choice
In quantum information science, matrix product states (MPS) and matrix product operators (MPO) are essential tools for efficiently representing quantum states and operations, particularly in simulations of quantum circuits such as the quantum Fourier transform (QFT). The efficiency and accuracy of these representations depend critically on parameters like bond dimension and error bounds.
For an MPS representation of the function f2(x) = sum of 20 cosines with n = 12, which of the following bond dimension sequences accurately describes how the required resources vary across the chain?
1) [2, 3, 5, 6, 7, 8, 10, 12, 7, 5, 2]
2) [2, 4, 7, 8, 8, 9, 10, 10, 8, 4, 2]
3) [2, 4, 6, 8, 9, 11, 13, 8, 4, 2, 2]
4) [2, 4, 6, 7, 8, 9, 11, 13, 8, 4, 2]
5) [2, 3, 6, 8, 10, 11, 12, 12, 9, 5, 2]
6) [3, 5, 7, 8, 9, 10, 12, 13, 9, 5, 3]
7) [2, 4, 5, 7, 8, 8, 9, 11, 7, 4, 2]
✓ Correct Answer:
The correct answer is 4) [2, 4, 6, 7, 8, 9, 11, 13, 8, 4, 2].
📚 Reference Text:
˜emax on both Dχ n+1,jand Dχ n+1,j+1; therefore ˜emaxonDχ n+1,jand Dχ n+1,j+1will also share the same bound, which makes it independent of j. For details, see the diagrammatic illus- tration in Fig. 9. Therefore, because of the decay of ˜σk j, the maximal error is also approximately O(e−χlog(χ/3)/√χ), and is thus at most O(ne−χlog(χ/3)/√χ)among the entire QFT MPO. APPENDIX K: FUNCTIONS USED FOR TIMING OF SUPERFAST FOURIER TRANSFORM For the timing of the superfast Fourier transform algorithm based on the MPO representation of the QFT, we used three example functions, all defined on the domain 0≤x≤1. The first function, labeled “one cosine” in Fig. 4,i s given by f1(x)=cos(2πx). (K1) It compresses exactly into an MPS with uniform bond dimensions all of size χm=2. The second function, labeled “20 cosines” in Fig. 4,i s given by f2(x)=20/summationdisplay j=1cos/bracketleftbigg 1.1(4j−2)/parenleftbigg x−1 2/parenrightbigg/bracketrightbigg . (K2) When this function was represented as an MPS with n=12, for example, the bond dimensions were [2, 4, 6, 7, 8, 9, 11, 13, 8, 4, 2]. The third function, labeled “one cosine + cusps” in Fig.4, is given by f3(x)=cos(2πx)+2e−3|x−0.4|+e−2|x−0.7| +2e−3|x−0.8|+e−2|x−0.9|. (K3) When this function was represented as an MPS with n=12, for example, the bond dimensions were [2, 4, 7, 8, 8, 9, 10, 10, 8, 4, 2]. [1] P. W. Shor, Polynomial-time algorithms for prime factor- ization and discrete logarithms on a quantum computer,SIAM J. Comput. 26, 1484 (1997). [2] A. Y. Kitaev, Quantum measurements and the Abelian sta- bilizer problem, Electron. Colloquium Comput. Complex. TR96 (1996). [3] P. Høyer, Conjugated operators in quantum algorithms, P h y s .R e v .A 59, 3280 (1999). 040318-29 JIELUN CHEN, STOUDENMIRE, and WHITE PRX QUANTUM 4,040318 (2023) [4] I. Kassal, J. D. Whitfield, A. Perdomo-Ortiz, M.-H. Yung, and A. Aspuru-Guzik, Simulating chemistry using quantumcomputers, Annu. Rev. Phys. Chem. 62, 185 (2011). [5] A. W. Harrow, A. Hassidim, and S. Lloyd, Quantum algorithm for linear systems of equations, P h y s .R e v .L e t t . 103, 150502 (2009). [6] L. Ruiz-Perez and J. C. Garcia-Escartin, Quantum arith- metic with the quantum Fourier transform, Quantum Inf. Process. 16, 152 (2017). [7] M. A. Nielsen, C. M. Dawson, J. L. Dodd, A. Gilchrist, D .M o r t i m e r ,T .J .O s b o r n e ,M .J .B r e m n e r ,A .W .H a r r o w ,and A. Hines, Quantum dynamics as a physical resource, Phys. Rev. A 67, 052301 (2003). [8] J. Tyson, Operator-Schmidt decomposition of the quantum Fourier transform on C N1tensor CN2,J. Phys. A: Math. General 36, 6813 (2003). [9] K. Woolfe, C. Hill, and L. Hollenberg, Scaling and effi- cient classical simulation of the quantum Fourier transform, Quantum Inf. Comput. 17, 1 (2014). [10] D. Slepian and H. O. Pollak, Prolate spheroidal wave func- tions, Fourier analysis and uncertainty—I, Bell Syst. Tech. J.40, 43 (1961). [11] H. J. Landau and H. O. Pollak, Prolate spheroidal wave functions,
Question 145multiple-choice
Quantum photonic systems rely on precise control of photons across multiple modes to perform computational and networking tasks. The combination of linear optical elements and specialized gates enables a wide range of quantum operations, though scalability and error mitigation remain key challenges.
In a quantum optical system with n indistinguishable photons distributed over m modes, what is the exact dimension of the Hilbert space describing possible photon configurations?
1) n × m
2) mⁿ
3) C(n + m - 1, n)
4) n! × m!
5) 2ⁿ
6) C(m, n)
7) n + m
✓ Correct Answer:
The correct answer is 3) C(n + m - 1, n).
📚 Reference Text:
[8] to prove that augmenting the linear optical unitaries with any nontrivial SNAP gate is sufficient to achieve universality. These results are in Sec. IV. • In Sec. V, a brief overview of two future directions related to this work. First we discuss the applicability to larger systems of our analysis techniques and results regarding distinguishability errors. We propose a low-order approx- imation of the effect of distinguishability errors, as elaborated upon in Remark 5. We then consider the problem of 3 how many SNAP gates are required to promote linear optics to an approximate 2-design and provide some numerical results for small system sizes. II. DISTINGUISHABILITY AND LOSS ERRORS Loss and distinguishability are two of the most dominant contributions to noise and errors in optical settings, both of which are known to reduce the classical complexity of computational tasks such as Boson sampling [9, 10]. As such, it is important to be able simulate the effects of each in software. Our starting point is Fock basis simulation. Generally we will be interested in simulating an mmode system with a fixed number nof (for now) identical photons. As such the Hilbert space can be represented via the second quantization as H= span {|n1, . . . , n m⟩}P ini=n, with dimension dn,m:= n+m−1 n . The initial state is typically composed of nsingle photons in m≥nmodes, |1⟩⊗n|0⟩⊗(m−n). Evolution occurs via a linear optical network, which maps creation operators1 to a linear combination Ua† iU†=mX j=1ujia† j, (1) where uis an m×munitary matrix, called the transfer matrix. It is known that any linear optical transformation can be implemented, to arbitrary accuracy, by a sequence of O(m2)beamsplitters2[12]. The full state output of such a network can be computed in time O(ndn,m), and space O(dn,m)[5, 13], by multiplying the evolved creation operators: U|1⟩⊗n|0⟩⊗(m−n)=UnY i=1a† iU†|⃗0⟩=nY i=1 mX j=1ujia† j |⃗0⟩. (2) In the first step we write |1⟩=a†|0⟩, and use that any linear optical unitary acting on the vacuum is the identity, i.e. U†|⃗0⟩=|⃗0⟩. In the second step we insert resolutions of the identity I=U†Uafter each creation operator. In this work we will assume measurements are all of the photon-number-resolving (PNR) type, which, in the absence of error, counts the number of photons in each measured mode. That is, it is equivalent to performing a measurement in the Fock basis. In Secs. II A and II B, we will discuss how to modify such a simulation to include losses and distinguishability errors. This is followed by Sec. II C, in which we review work of Marshall [6] that gives a protocol for distilling less distinguishable photons. This protocol was discovered using the simulation techniques of Sec. II B. Then in Sec. II D, we consider the effect that distinguishability has on circuits for fusion, GHZ state generation, and related operations. The first several subsections give analytical results; Sec. II D 3 gives numerics for Bell state generation in the presence of distinguishability, using the techniques of Sec. II B. A. Simulation of Loss
Question 146multiple-choice
Quantum algorithms have revolutionized certain computational tasks, particularly those involving algebraic structures and hidden patterns in functions over finite fields. Understanding the difference in complexity between classical and quantum approaches to identifying hidden polynomials is essential in computational algebra and cryptography.
When identifying an m-variate polynomial of total degree greater than 2 hidden within a function over a finite field with d elements, which of the following statements best describes the quantum algorithm's advantage over classical algorithms?
1) The quantum algorithm requires exponentially more queries than the classical algorithm as d increases.
2) Both quantum and classical algorithms require Ω(d) queries for constant success probability.
3) Quantum algorithms do not offer any speedup for polynomials of degree greater than 2.
4) The classical algorithm is faster due to its ability to directly sample all polynomial coefficients.
5) Quantum algorithms cannot identify polynomials of degree greater than 2 with constant probability.
6) The quantum algorithm identifies hidden polynomials with constant probability in polylogarithmic time, whereas classical methods require at least Ω(√d) queries.
7) Classical algorithms outperform quantum algorithms when d is large and m > 1.
✓ Correct Answer:
The correct answer is 6) The quantum algorithm identifies hidden polynomials with constant probability in polylogarithmic time, whereas classical methods require at least Ω(√d) queries..
📚 Reference Text:
Title: Efficient quantum algorithm for identifying hidden polynomials Year: 2007 Paper ID: ae9fac4d6a153f60743cb57025bb83cc643f49bf Source: semantic-scholar URL: https://www.semanticscholar.org/paper/ae9fac4d6a153f60743cb57025bb83cc643f49bf Abstract: We consider a natural generalization of an abelian Hidden Subgroup Problem where thesubgroups and their cosets correspond to graphs of linear functions over a finite field Fwith d elements. The hidden functions of the generalized problem are not restricted tobe linear but can also be m-variate polynomial functions of total degree n > 2.The problem of identifying hidden m-variate polynomials of degree less or equalto n for fixed n and m is hard on a classical computer since Ω(√d) black-box queriesare required to guarantee a constant success probability. In contrast, we present aquantum algorithm that correctly identifies such hidden polynomials for all but a finitenumber of values of d with constant probability and that has a running time that is onlypolylogarithmic in d.
Question 147multiple-choice
Quantum algorithms for hidden subgroup problems use group-theoretic methods to efficiently uncover hidden structure in functions defined over groups. The complexity and implementation of the quantum Fourier transform vary significantly between abelian and non-abelian groups due to differences in their representations.
In the context of quantum algorithms for hidden subgroup problems, which of the following is a direct consequence of the structure of non-abelian groups when implementing the quantum Fourier transform?
1) The Fourier basis states are always equal to the standard basis states.
2) All irreducible representations are one-dimensional characters.
3) The quantum algorithm requires an exponential number of queries even for simple groups.
4) The irreducible representations used to construct the Fourier basis are higher-dimensional unitary matrices.
5) The Fourier transform over non-abelian groups can be implemented using only Hadamard gates.
6) The sum of the dimensions of irreducible representations equals the group order.
7) The quantum Fourier transform does not require orthogonality of basis states.
✓ Correct Answer:
The correct answer is 4) The irreducible representations used to construct the Fourier basis are higher-dimensional unitary matrices..
📚 Reference Text:
but to convert this into an explicit description of K(say an actual set of generators) we need to use further mathematical properties of Ge.g. properties of co-primality as illustrated in our examples. The above group-theoretic framework serves to generalize a nd extend the applicability of the quantum algorithm for periodicity determination. Fo r example Simon considered the following problem: suppose that we have a black box which com putes a function ffromn-bit strings ton-bit strings. It is also promised that the function is “two-t o-one” in the sense that there is a fixed n-bit stringξsuch that f(x+ξ) =f(x) for all n-bit strings x. (18) (Here + denotes binary bitwise addition of nbit strings.) Our problem is to determine ξ. To see that this is just a generalized periodicity determina tion note that in the group ( Z2)n ofn-bit strings, every element satisfies x+x= 0. Hence eq. (18) states just that fis periodic on the group with periodicity subgroup K={0,ξ}. Thus to determine ξwe construct the 10 Fourier transform on the group of n-bit strings and apply the standard algorithm above. The relevant Hilbert space Hwith a basis labeled by n-bit strings is just a row of nqubits. The irreducible representations of the group ZN 2are the functions fx(y) = (−1)x1y1...(−1)xnyn wherex=x1...x nandy=y1...y nare n bit strings. Thus the Fourier transform may be easily seen [11] to be just the application of the 1-qubit Had amard transform: H=1√ 2/parenleftBigg 1 1 1−1/parenrightBigg to each of the nqubits. The resulting quantum algorithm for determining th e hidden subgroup then reproduces Simon’s original algorithm [6]. It determi nesξinO(n2) steps whereas it may be argued [6] that any classical algorithm must evaluate fat leastO(2n) times. 6 Non-abelian groups We will now consider the hidden subgroup problem in the situa tion whereGand the subgroup Kmay be non-abelian i.e. we have f:G→Xwhich is constant on the (left) cosets of Kin G. We now also write the group operation multiplicatively. As before our algorithm begins in the same way by producing the state |g0K/an}bracketri}htwhereg0has been chosen at random. The passage from abelian to non-abelian groups is accompanied b y various potential conceptual problems: (a) (Construction of non-abelian Fourier transform). For a belian groups the irreducible representations are always one dimensional (i.e. the funct ionsχlin eq. (17)) whereas for non-abelian groups they are functions χ:G→U(d) taking values in the set U(d) of all d×dunitary matrices for suitable values of d. According to a basic theorem of group representation theory [3], if d1...,d mare the dimensions of a complete set of irreducible unitary representations χ1,...,χ mthend2 1+...+d2 m=|G|. Let us write χi,jk(g) for the (j,k)th component of the unitary matrix χi(g). Thus asi,j,kvary we get |G|complex valued functions and as in eq. (17) we may define the |G|states: |χi,jk/an}bracketri}ht=1/radicalbig |G|/summationdisplay g∈Gχi,jk(g)|g/an}bracketri}ht. The orthogonality relations of irreducible representatio ns [3] guarantee that these are again orthonormal states, called the Fourier basis, and the non-a belian Fourier transform is defined as the unitary operation that rotates this basis into standa rd
Question 148multiple-choice
Quantum algorithms have shown significant speedups for certain algebraic problems, but their performance often depends on the underlying algebraic structure. The complexity of membership and discrete logarithm problems varies greatly between groups and semigroups.
Which statement correctly characterizes the quantum query complexity of the constructive membership problem in general semigroups with k ≥ 2 generators?
1) It can always be solved with a polynomial number of quantum queries regardless of the semigroup structure.
2) It requires only logarithmic quantum queries in the size of the semigroup.
3) It requires exponentially many quantum queries for general semigroups when there are at least two generators.
4) It is efficiently solvable by Shor’s algorithm for any semigroup.
5) It becomes trivial if the semigroup is non-abelian.
6) It does not depend on the number of generators.
7) It is equivalent in difficulty to the discrete logarithm problem in abelian groups.
✓ Correct Answer:
The correct answer is 3) It requires exponentially many quantum queries for general semigroups when there are at least two generators..
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solve the shifted discrete logarithm problem in a black-box semigroup Swith only poly .logjSj/queries. As in the proof of Theorem 3.1, given a candidate value a, we can check whetherxDyga. If this check fails, we can conclude (with bounded error) that no solution exists. The dihedral hidden subgroup problem (DHSP) is apparently hard. Despite con- siderable effort (motivated by a close connection to lattice problems [17]), Kuper- berg’s algorithm remains the best known approach, and it is plausible that there might be no efficient quantum algorithm. Note that the DHSP can be reduced to a quantum generalization of the constructive orbit membership problem, namely, Quantum computation of discrete logarithms in semigroups 411 orbit membership for a permutation action on pairwise orthogonal quantum states [7, Proposition 2.2]. Thus, intuitively, a solution of the shift problem for a (classi- cal) permutation action (such as in the shifted discrete logarithm problem) should exploit that the action is on classical states, unless it also solves the DHSP. In Section 5, we describe another variant of the discrete logarithm problem that is even harder than the shifted discrete logarithm problem, requiring exponentially many queries. We also show that our lower bound for that problem is nearly opti- mal using the algorithm of Lemma 4.1 as a subroutine. 5 Constructive semigroup membership Given an abelian semigroup generated by g1;:::;gkand a semigroup element x2 hg1;:::;gki, the constructive membership problem asks us to find a1;:::;ak2 N0´¹0;1;2;:::ºwitha1CCak1such thatxDga1 1gak k. The notation g0 isimply indicates that no factor of giis present, so solutions with aiD0for some values of iare well defined even though the semigroup need not have an identity element. This natural generalization of the discrete logarithm problem is easy for abelian groups (see for example [10, Theorem 5]). In that case, let ri´jhgiijfor alli2 ¹1;:::;kº,r´jhxij, andL´Zr1 ZrkZr. The values.r1;:::;rk;r/ can be computed efficiently by Shor’s order-finding algorithm [19]. Now consider the functionfWL!Gdefined byf.a1;:::;ak;b/Dga1 1gak kx b. This func- tion hides the subgroup H´® .a1;:::;ak;b/2LWga1 1gak kDxb¯ L; so generators of Hcan be found in polynomial time [15]. To solve the construc- tive membership problem, it suffices to find the solutions with bD1modr. This corresponds to a system of linear Diophantine equations, so it can be solved clas- sically in polynomial time (see for example [18, Corollary 5.3b]). Here we show that the constructive membership problem in semigroups is con- siderably harder. Specifically, given a black-box semigroup S, we need expo- nentially many quantum queries (in log jSj) to solve the constructive membership problem with respect to k2generators. Theorem 5.1. For any fixed k2N, there is a black-box semigroup Swithk generators for which at least .jSj1 2 1 2k/quantum queries are required to solve the constructive membership problem. Proof. For anyn2N, consider the abelian semigroup SD® ga1 1gak kWa1;:::;ak2N0;1a1CCakn¯ [¹0º 412 A. M. Childs and G. Ivanyos generated by g1;:::;gk, with the following multiplication rules: 0.ga1 1gak k/D0; .ga1 1gak k/.gb1 1gbk k/D´ ga1Cb1 1gakCbk kifPk iD1.aiCbi/n, 0 ifPk iD1.aiCbi/>n: Let´¹.a1;:::;ak 1/2Nk 1 0Wa1CCak 1nº. We show that the problem of inverting a
Question 149multiple-choice
Quantum information theory studies how information can be transmitted and processed using quantum systems, with special attention to the effects of channel noise and symmetries. Entanglement-assisted classical capacity quantifies the optimal rate for sending classical bits over quantum channels when unlimited entanglement is shared between sender and receiver.
Which statement correctly describes the impact of a quantum Markov semigroup satisfying a modified logarithmic Sobolev inequality (MLSI) on the entanglement-assisted classical capacity of quantum channels with finite symmetry groups?
1) MLSI causes the capacity to remain constant over time regardless of channel evolution.
2) MLSI guarantees polynomial decay of the capacity upper bound as time increases.
3) MLSI leads to exponential decay of the relative entropy, resulting in exponentially tightening upper bounds for capacity over time.
4) MLSI only affects capacities in infinite-dimensional Hilbert spaces, not finite ones.
5) MLSI increases the channel capacity by enhancing entanglement generation.
6) MLSI eliminates the strong converse property for these channels.
7) MLSI ensures that lower bounds on capacity exceed upper bounds for all time.
✓ Correct Answer:
The correct answer is 3) MLSI leads to exponential decay of the relative entropy, resulting in exponentially tightening upper bounds for capacity over time..
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/uni27E9/uni27E8 /divides.alt0) on systemBand where /divides.alt0 /uni27E9/uni27E8 /divides.alt0stands for the rank-one orthogonal projection on the norm-one vector ∈H. A closed formula for the entanglement assisted capacity 1note that we are just rewriting the mutual information in terms of the relative entropy in (56).was obtained in [34]. There, they show that for a conditional expectation Efixwe have CEA(Efix)=log/parenleft.alt4m /summation.disp i=1n2 i/parenright.alt4: Using similar ideas as before we can estimate the entanglement assisted classical capacity using relative entropy transference. Theorem V .6 (Bounding the entanglement assisted classical capacity) .Let(Tt)t≥0be a transferred QMS. Then: log/parenleft.alt4m /summation.disp i=1n2 i/parenright.alt4≤CEA(Tt())≤log/parenleft.alt4m /summation.disp i=1n2 i/parenright.alt4+: Furthermore, if (St)t≥0satisfies an -MLSI and Gis finite, then: CEA(Tt)≤log/parenleft.alt4m /summation.disp i=1n2 i/parenright.alt4+e−2 tlog(/divides.alt0G/divides.alt0) Moreover, the upper bounds are in the strong converse sense. Proof. The proof is similar to the ones of Theorems V .1 and V .5 and Proposition V .13, and hence is omitted. D. Capacities from a modified logarithmic Sobolev inequality Similarly to the derivation of decoherence times, one expects to get tighter bounds on the various capacities by directly applying a quantum functional inequality, when the latter is known. As was the case with decoherence times, the decay of the capacities we obtain does not depend on particular properties of the representation at hand and will in general not be tight. For capacities, the right functional inequality to consider is the modified logarithmic Sobolev inequality (MLSI). To the best of our knowledge, this connection between a MLSI and capacity bounds cannot be found in the literature beyond primitive semigroups [46], so we establish it here for more general semigroups. Here, we still assume that (Tt)t≥0is a quantum Markov semigroup on B(H)that is symmetric with respect to the Hilbert Schmidt inner product. Then, instead of using the entropy comparison theorem (Theorem III.1), one can simply decompose the relative entropy between Tt(), ∈D(H), and any state ∈D(Nfix)as follows: Lemma V .7. For any∈D(H), and any∈D(Nfix), D(Tt()/parallel.alt1)=D(Efix()/parallel.alt1)+D(Tt()/parallel.alt1Efix()): 18 Proof. D(Tt()/parallel.alt1) =Tr(Tt()(ln(Tt())−ln)) =Tr(Tt()(ln(Tt())−ln(Efix()))) +Tr(Tt()(ln(Efix())−ln)) =D(Tt()/parallel.alt1Efix())+Tr(Efix()(ln(Efix())−ln)) =D(Tt()/parallel.alt1Efix())+D(Efix()/parallel.alt1); where in the third line we used that Efixis a conditional expectation with respect to the completely mixed state, so thatEfix=E† fixand for any ∈D(Nfix)∩D(H)+, ln()=Efix(ln()). We recall that, given a faithful quantum Markov semigroup (Tt=e−tL)t≥0, its decoherence-free modified logarithmic Sobolev constant 1(L)has been defined in [27] as follows: 1(L)∶=inf ∈D+(H)Tr(L()(ln−lnEfix())) D(/parallel.alt1Efix()) (we recall our convention that Lis a positive semi-definite operator). It is the largest constant >0 such that the following decay in relative entropy occurs2: D(Tt()/parallel.alt1Efix())≤e− tD(/parallel.alt1Efix()): The classical logarithmic Sobolev inequality is then defined in an analogous way. An extension of Theorem III.1 then gives the Theorem V .8. Let(Tt)t≥0be a transferred QMS from a finite group GonB(H),∈D(Nfix)and suppose that Sthas an MLS constant 1(L)>0. Then, for any state ∈D(H)andt≥0: D(Efix()/parallel.alt1)≤D(Tt()/parallel.alt1) (57) ≤D(Efix()/parallel.alt1)+e− 1(L)tlog(/divides.alt0G/divides.alt0) Proof. It follows from Theorem III.1 that D(Tt()/parallel.alt1)≤D(Efix()/parallel.alt1)+/integral.dispGktlog(kt)dG: Now note that /integral.dispGktlog(kt)dG=D(t/parallel.alt1G); wheregis an arbitrary element of the group and where dt=kt∗gdG. Remark also that t=St(g). Thus, as we assumed that Stsatisfies a MLSI, we conclude that D(t/parallel.alt1G)≤e− 1(L)tlog(/divides.alt0G/divides.alt0); (58) where in the last step we used the
Question 150multiple-choice
Parity encoding schemes in quantum computing architectures offer alternative ways to represent logical qubits and perform universal gate operations. The Lechner-Hauke-Zoller (LHZ) architecture leverages parity qubits and data qubits to address hardware connectivity challenges and optimize quantum algorithm implementation.
Within the LHZ architecture, which operation specifically requires chains of CNOT gates due to the structure of its corresponding logical operator, making its implementation more complex than single-qubit rotations?
1) Logical Y rotation (˜Ry)
2) Logical X rotation (˜Rx)
3) Logical Z rotation (˜Rz)
4) Physical Z rotation on a single data qubit
5) Measurement in the computational basis
6) Preparation of graph states
7) Application of a single Hadamard gate
✓ Correct Answer:
The correct answer is 2) Logical X rotation (˜Rx).
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zero or an even number oftimes. The square brackets around σ (l4) zindicate that a con- straint can contain either three or four qubits. The specialcase of the parity encoding involving all n(n−1)/2 two-body terms (parity qubits) for nlogical qubits is known as the LHZ architecture [ 12]. It is possible to extend this by including physical qubits representing single logical qubits such that ˜σ (i) z=σ(i) z. (3) We refer to these qubits as data qubits in the following. The presence of data qubits in the parity encoding ensures that thephysical qubits uniquely define the state of the logical qubits. 1 A variant of the LHZ architecture involving all data qubits hasrecently been shown to provide a universal gate set [ 11], based on the logical operators ˜R (i) x(α)=exp/parenleftBigg −iα 2σ(i) x/productdisplay jiσ(ij) x/parenrightBigg , (4) ˜R(i) z(α)=exp/parenleftBig −iα 2σ(i) z/parenrightBig =R(i) z(α), (5) C˜P(i,j) φ=R(i) z/parenleftbiggφ 2/parenrightbigg R(ij) z/parenleftbigg −φ 2/parenrightbigg R(j) z/parenleftbiggφ 2/parenrightbigg . (6) 1Note that two-body parity qubits alone determine the logical state only up to a global spin flip. 2469-9926/2022/106(4)/042442(8) 042442-1 Published by the American Physical Society FELLNER, MESSINGER, ENDER, AND LECHNER PHYSICAL REVIEW A 106, 042442 (2022) Logical operators (with a tilde) act on the logical qubits defined in the constraint-fulfilling subspace and commutewith all constraint operators. In contrast to that, physicaloperators (without a tilde) do not necessarily preserve theconstraint-fulfilling subspace. The ˜R xoperators require chains of controlled- NOT (CNOT ) gates due to the product of Pauli operators in the exponent (see, for example, Refs. [ 30–32]f o r further background on exponentiating products of Pauli ma-trices), while the other logical operators can be implementedwith physical single-qubit operations only. The set of physicalqubits that are involved in ˜R (i) x(i.e., the set of all physical qubits containing the logical index i) is referred to as the logical line associated with qubit ( i). In this work we study the implementation of several es- sential quantum algorithms in this scheme and find that,depending on the algorithm, the parity scheme can show anadvantage in circuit depth or multiqubit gate count. In par-ticular, we focus on quantum algorithms essential for Shor’sfactoring algorithm [ 2], the quantum Fourier transform (QFT) [31], and the quantum addition algorithm based on the QFT [33], as well as the implementation of Grover’s diffusion operator [ 4]. Furthermore, we present a strategy to efficiently prepare graph states, which represent an important resourcefor measurement-based quantum computing [ 34]. II. COMMON GATES AND GATE SEQUENCES A. Arbitrary single-qubit gates Any single-qubit unitary Ucan be decomposed into rota- tions [ 31] U=Rz(α)Rx(β)Rz(γ), (7) with some angles α,β, andγ. We can thus construct any log- ical single-qubit gate using the operators defined in Eqs. ( 4) and ( 5)a s ˜U=˜Rz(α)˜Rx(β)˜Rz(γ). (8) The two ˜Rzrotations can be easily implemented in the LHZ scheme with physical rotations on the corresponding dataqubits. The ˜R xrotation requires a chain of CNOT gates along the logical line and a physical Rxrotation on one of its qubits,
Question 151multiple-choice
Quantum algorithms leverage specialized circuit operations to efficiently manipulate and measure properties of quantum states, particularly in problems involving symmetry and signal processing. Techniques such as the quantum Fourier transform (QFT), phase gates, and the Hadamard test are foundational for tasks like operator diagonalization and expectation value estimation.
Which circuit technique enables efficient diagonalization of the cyclic permutation operator in quantum computing, resulting in circuit depth that does not scale with the exponent of the permutation?
1) Controlled-X gate sequences combined with classical shift operations
2) Grover diffusion operator implemented repeatedly
3) Use of Toffoli gates for iterative cyclic shifts
4) Variational quantum eigensolver with parameterized rotation gates
5) Classical fast Fourier transform applied before measurement
6) Quantum Fourier Transform (QFT) followed by phase gates
7) Quantum error correction codes integrated with permutation logic
✓ Correct Answer:
The correct answer is 6) Quantum Fourier Transform (QFT) followed by phase gates.
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1 with quantum computers, obtaining estimations of each entry of VR,VI, qR, and qIwith the Hadamard test [22], which are then stored in classical memory. We modify quantum circuit implementations of shift operations shown by Koch et al. [30] and quantum adder circuit by Draper [31] to implement the cyclic permutation operator Qand its powers with circuit depth independent to the power m, a contrast to the implementation of quantum circuits for solving general linear systems by Huang et al. [7]. Steps 5-8 of the algorithm remain implemented on classical computers. As shown in the following section, the powers of Qcan be implemented with a known quantum circuit with O(n2) quantum gates. Therefore, we can construct classical sketches of the circuit implementation of output states |um⟩=QmUb|0n⟩that correspond to the computed optimal parameters αm, using O(Tn2) memory in total to provide a classical sketch of the estimator ˜ xas a combination of quantum states. We now provide details of the circuit implementation as follows: Quantum circuit implementation. Recall from Section II that the eigendecomposition of the cyclic permutation matrix Qis given by Q=F−1ΛF, where Λ is the diagonal matrix with all eigenvalues of Qand the diagonalization matrix Fis the matrix representation of DFT. On quantum systems, for cases where n= log2(N),Fcan be implemented with O(n2) quantum logic gates using the quantum Fourier transform (QFT) [17]. For the more general case where n̸= log2N, one can use the arbitrary-size quantum Fourier transform proposed by Kitaev [32] which utilizes quantum phase estimation to construct DFT. As such implementations require fault-tolerance, we discuss only the standard QFT for our implementation, while details on the generalized case are provided in Appendix B. 7 (a) ...Λ =P(π) P(π 2) P(2π N) (b) ...Λ =P(mπ) P(mπ 2) P(2mπ N) (c) |0⟩H I/S H |0⟩ UbQFT Λm ... |0⟩ FIG. 2. Circuit implementation of quantum subroutines in our algorithm. The cyclic permutation matrix Qcan be diago- nalized by the DFT matrix Fsuch that Q=F−1ΛF. DFT can be implemented by QFT circuits. See Appendix B for the implementation of QFT for arbitrary N. The diagonal matrix Λ can be implemented as a tensor product of phase gates as shown in Panel 2a. Powers of Λ can be implemented with the same phase gates, by multiplied rotation angles as shown in Panel 2b. To retrieve the real and imaginary parts of ⟨b|Qm|b⟩, we use the Hadamard test (on the state QFT|b⟩), which requires a controlled version of Λm, i.e. controlled phase gates, as shown in Panel 2c. For the remainder of the discussion, we assume n= log2(N), while keeping in mind that cases where n̸= log2(N) are still implementable. Matrix Λ can be written as: Λ = diag ω0 Nω1 Nω2 N···ωN−1 N (14) We use the phase gate P, whose matrix form is P(θ) =1 0 0eiθ , (15) where the parameter θis the rotation angle. We note that ω0= 1. Hence Λ can be written as the tensor product of P(θ) gates as follows: Λ =1 0 0ωN/2
Question 152multiple-choice
Quantum magnetism and many-body physics often require specialized computational techniques and careful handling of symmetries to explore exotic phases such as spin liquids and quantum paramagnets. Theoretical frameworks like Schur-Weyl duality and advanced variational ansatzes are essential for efficient simulation and understanding of these systems.
Which property distinguishes the eSWAP ansatz from the Sn-CQA ansatz in preserving global SU(2) symmetry for spin systems, but not extending its universality to higher-dimensional (qudit) cases?
1) eSWAP supports adiabatic evolution at large circuit depth with global SU(d) symmetry
2) eSWAP universally preserves SU(2) symmetry in relevant sectors, but loses universality for qudits
3) eSWAP alternates between problem and mixer Hamiltonians, resembling QAOA
4) eSWAP is commutative and requires fixed circuit layout
5) eSWAP enables efficient initialization using the Young basis for all local dimensions
6) eSWAP guarantees ground state location in frustrated systems via the Marshall-Lieb-Mattis theorem
7) eSWAP performs exact diagonalization for large-scale quantum systems
✓ Correct Answer:
The correct answer is 2) eSWAP universally preserves SU(2) symmetry in relevant sectors, but loses universality for qudits.
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is known about the intermediate quantum paramagnetic phase—recent evidence of decon- fined quantum criticality [ 84,85] sparked further interest in studying these regimes. Gaining physical insights in the intermediate quantum paramagnetic phase requires solving the problem of the ground-state sign structure the system approaches the phase transition. Recently, there were a number of numerical attempts to address the existence of the U(1)gapless spin-liquid phase, using recently the ten- sor networks [ 86], restricted Boltzmann machine (RBM) [87], convolutional neural network (CNN) [ 57,58,88], and graphical neural network (GNN) [ 89]—all yielding partial progress. As a significant difference from the unfrustrated case, the Marshall-Lieb-Mattis theorem does not hold gen- erally and there is no guarantee that the ground state still lives at J=0 or equivalently λ=(n/2,n/2), which urges us to preserve the global SU(2) symmetry, which fur- ther gives us access to search in all inequivalent Snirreps decomposed from the system by Schur-Weyl duality. Global SU (2)symmetry and challenges in NQS ansätze Taking advantage of the global SU (2)symmetry, we address this problem in a different way: we recast the Hamiltonian in Eq. (8)by the following identity: π((ij))=2ˆSi·ˆSj+1 2I,( 9 ) withˆSibeing further expanded as the half of standard Pauli operators {X,Y,Z}. Equation (9)was first discov- ered by Heisenberg himself [ 90,91] (an elementary proof can be found in the Appendix) and more recently noted by Ref. [ 46] in analyzing the ground-state property of the 1D Heisenberg chain. As designed by products of expo- nentials of SWAPs (eSWAPs), the method proposed inRef. [ 46] truly preserves the global SU(2) symmetry. As a brief comparison with S n-CQA, eSWAP ansätze are uni- versal in relevant sectors given by the SU(2) symmetry. However, this property no longer holds for qudits with 020327-12 SPEEDING UP LEARNING QUANTUM... PRX QUANTUM 4,020327 (2023) FIG. 9. Scaling properties of small total spin irreps dimen- sion, respectively. The graph shows the scale: 2n/dimSλwith the partitions λ1=(n/2,n/2)(red),λ2=(n/2+1,n/2−1)(gray), λ3=(n/2+2,n/2−2)(orange). The orange dashed line is exp(−n/8)the exponential decay, since the plot starts at n=8. d≥3 (Sec. III C). As eSWAPs are noncommutative oper- ators in general, there are various ways to place them in a quantum circuit. A more suitable perspective to describe the ansatz might be sampling them as two-local SU(2) ran- dom circuits [ 45]. On the other hand, the Sn-CQA ansatz is designed by alternating exponentials of the problem and mixer Hamiltonians HP,HMjust like the framework of QAOA. Similar to QAOA, Sn-CQA at large pcorre- sponds to a form of adiabatic evolution with global SU (d) symmetry, which could hint a theoretically guaranteed performance as pis large (see Sec. VII in the Appendix). An immediate consequence of using Eq. (9)is that the resulting Heisenberg Hamiltonian can be expressed in the Young basis where every Snirrep is indexed by the total spin label j. Mapping to this basis can be done using the constant-depth circuit state initialization in Sec. IV B . Using our C[Sn] variational ansatz leads to a more effi- cient algorithm by polynomially reducing the
Question 153multiple-choice
Quantum computing offers novel approaches for solving structured linear systems, particularly those with circulant and banded matrix properties, which frequently arise in physics and engineering applications. Efficient algorithms for these systems can leverage matrix symmetries and specialized quantum operations to reduce computational resources.
Which algorithmic strategy enables a reduction from exponential to linear quantum resource requirements when solving banded circulant linear systems with bandwidth parameter \( K \)?
1) Using Grover’s search to optimize eigenvalue estimation
2) Encoding matrix entries directly into quantum amplitudes
3) Employing classical iterative refinement after quantum subroutines
4) Applying amplitude amplification to classical solutions
5) Decomposing banded circulant matrices into cyclic permutations and combining quantum states via convex optimization
6) Utilizing quantum phase estimation on non-circulant matrices
7) Implementing variational quantum eigensolvers without matrix decomposition
✓ Correct Answer:
The correct answer is 5) Decomposing banded circulant matrices into cyclic permutations and combining quantum states via convex optimization.
📚 Reference Text:
Title: Hybrid quantum-classical and quantum-inspired classical algorithms for solving banded circulant linear systems Year: 2023 Paper ID: e80cfd7fa416b97298b23b624f92f8e58e975b56 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/e80cfd7fa416b97298b23b624f92f8e58e975b56 Abstract: Solving linear systems is of great importance in numerous fields. In particular, circulant systems are especially valuable for efficiently finding numerical solutions to physics-related differential equations. Current quantum algorithms like HHL or variational methods are either resource-intensive or may fail to find a solution. We present an efficient algorithm based on convex optimization of combinations of quantum states to solve for banded circulant linear systems whose non-zero terms are within distance $K$ of the main diagonal. By decomposing banded circulant matrices into cyclic permutations, our approach produces approximate solutions to such systems with a combination of quantum states linear to $K$, significantly improving over previous convergence guarantees, which require quantum states exponential to $K$. We propose a hybrid quantum-classical algorithm using the Hadamard test and the quantum Fourier transform as subroutines and show its PromiseBQP-hardness. Additionally, we introduce a quantum-inspired algorithm with similar performance given sample and query access. We validate our methods with classical simulations and actual IBM quantum computer implementation, showcasing their applicability for solving physical problems such as heat transfer.
Question 154multiple-choice
Quantum algorithms for linear algebra and simulation frequently exploit circuit decompositions and measurement strategies to optimize resource efficiency. Understanding the implementation and analysis of quantum operators is essential for evaluating algorithmic scalability and error bounds.
Which of the following statements best describes why the quantum circuit for implementing powers of the operator Λm achieves scalability regardless of the exponent m?
1) The circuit uses ancilla qubits to store the value of m, eliminating dependence on its size.
2) Λm is implemented via iterative Grover rotations, making the runtime independent of m.
3) The circuit employs amplitude amplification to compress the effect of m into a single operation.
4) The circuit relies on Hamiltonian simulation, where the time evolution parameter substitutes for m.
5) The operator Λm is realized using controlled-NOT gates whose number does not depend on m.
6) The implementation uses classical preprocessing to encode m into initial states, removing m from the circuit depth calculation.
7) The circuit’s depth is determined solely by the Quantum Fourier Transform, and the phase gate powers are set by multiplying phase angles, so the depth does not depend on m.
✓ Correct Answer:
The correct answer is 7) The circuit’s depth is determined solely by the Quantum Fourier Transform, and the phase gate powers are set by multiplying phase angles, so the depth does not depend on m..
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N ⊗ ··· ⊗ 1 0 0ω2 N ⊗ 1 0 0ωN =n−1O j=0P(θj),where θj=2π N·2j. (16) The implementation of Λ is then as illustrated in Figure 2a. We note that this is the non-controlled version of the quantum adder circuit proposed by Draper [31]. The number of gates for this implementation is equal to the qubit number n. We note that for m∈Z,Qmhas eigendecomposition F−1ΛmF. Further note that the powers of phase operations are phase operations with multiplied rotation angles, that is, P(θ)m=P(mθ). As Λ can be decomposed into a tensor product of phase gates, we can simply implement Λmwith the same gate, but with the angles multiplied by m, as shown in Figure 2b. Observe that the implemented quantum circuit is independent of mand only depends on the depth of the QFT implementation. Given the above implementation of powers of Q, to obtain the expectation value of ⟨b|Qm|b⟩, we prepare a quantum state|ψb⟩=QFT|b⟩. A Hadamard test is then performed on Λmto calculate the expectation values with respect to|ψb⟩as described in Appendix B, such that we obtain the values of Re ⟨ψb|Λm|ψb⟩and Im ⟨ψb|Λm|ψb⟩, which are equivalent to Re ⟨b|Qm|b⟩and Im ⟨b|Qm|b⟩. The full quantum circuit is shown in Figure 2c. Number of total measurements. As we conduct quantum measurements to obtain auxiliary system matrices Wand r, we do not gain the exact value of the entries in the matrices. Instead, what we obtain are estimations of the values whose accuracy depends on the number of measurements conducted. The total number of measurements required to achieve a close enough estimate of ˜ xis stated in the following proposition: Proposition 6 (Number of measurements needed) .Given K-banded circulant matrix C=PK ℓ=−KcℓQℓ∈ M N(C), hardware efficient implementation of Ubas shown in Assumption 5 and truncation threshold T. This defines auxiliary systems Wandrin Equation 12. Let the sum of absolute values of the coefficients of decomposed CbeB=PK ℓ=−K|cℓ|. We can find ˜α:{˜αm∈C,∀m∈[−T..T]}for estimator ˜x(α) =PT m=−TαmQmUb|0n⟩such that the following is satisfied ∥C˜x(˜α)− |b⟩∥2≤min α∈C2T+1∥C˜x(α)− |b⟩∥2+ε usingO(B4(K+T)T∥W∥∥W−1∥2(1 +∥W−1r∥2/ε)measurements via Hadamard test. 8 Algorithm 2 Classical subroutine for inner product estimation with sample and query access Input: Classical data structure Sbas shown in Assumption 8, Power of cyclic perturbation matrix m, Additive error ε, Failure rateδ Output: ε-close estimation of ⟨b, Qmb⟩with success rate 1 −δ 1:fori←0 to 6 log(2 /δ)do 2: ηi←0 3: forj←0 to 9 /ε2do 4: s←Sample (Sb) 5: q←Query (Sb, s−m(mod N)) 6: r←Query (Sb, s) 7: ηi←ηi+q/r 8: end for 9: ηi←ηiε2/9 10:end for 11:return Median (η) We note that the number of different expectation values needed is 4 K+ 4T+ 1∈ O(K+T) due to the high overlap of values required. So the number of total measurements for all entries on Wandrto have variance O(1/R) is O((K+T)B4R). From Proposition 12 of [7], we have R > CT ∥W∥∥W−1∥2(1 +∥W−1r∥2/ε). We now provide informal bounds of the above proposition. Based on discussions given by Huang et al. [7] on the upper bounds of ∥W∥,∥W−1∥and∥W−1r∥, we can see that the number
Question 155multiple-choice
Quantum computing hardware is limited by the number of available qubits and susceptibility to errors, making efficient circuit designs crucial for practical applications. The Quantum Fourier Transform (QFT) is a foundational algorithm, often used in tasks such as factoring, but typically requires significant resources.
Which property most directly enables an in-place quantum Fourier transform circuit to minimize hardware requirements while maintaining logarithmic depth?
1) Utilization of randomized measurements to reduce error rates
2) Addition of entangled ancilla qubits for error correction
3) Implementation of global gates acting on all qubits simultaneously
4) Encoding input states into higher-dimensional Hilbert spaces
5) Operating entirely without ancilla qubits and using only local gates in a 1D array
6) Repetition of QFT circuits with post-selection for accuracy
7) Reliance on adaptive classical feedback during computation
✓ Correct Answer:
The correct answer is 5) Operating entirely without ancilla qubits and using only local gates in a 1D array.
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Title: A log-depth in-place quantum Fourier transform that rarely needs ancillas Year: 2025 Paper ID: dd4413cc764e6b7f82f4b0e6c2bebb112c512e67 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/dd4413cc764e6b7f82f4b0e6c2bebb112c512e67 Abstract: When designing quantum circuits for a given unitary, it can be much cheaper to achieve a good approximation on most inputs than on all inputs. In this work we formalize this idea, and propose that such"optimistic quantum circuits"are often sufficient in the context of larger quantum algorithms. For the rare algorithm in which a subroutine needs to be a good approximation on all inputs, we provide a reduction which transforms optimistic circuits into general ones. Applying these ideas, we build an optimistic circuit for the in-place quantum Fourier transform (QFT). Our circuit has depth $O(\log (n / \epsilon))$ for tunable error parameter $\epsilon$, uses $n$ total qubits, i.e. no ancillas, is local for input qubits arranged in 1D, and is measurement-free. The circuit's error is bounded by $\epsilon$ on all input states except an $O(\epsilon)$-sized fraction of the Hilbert space. The circuit is also rather simple and thus may be practically useful. Combined with recent QFT-based fast arithmetic constructions [arXiv:2403.18006], the optimistic QFT yields factoring circuits of nearly linear depth using only $2n + O(n/\log n)$ total qubits. Additionally, we apply our reduction technique to yield an approximate QFT with well-controlled error on all inputs; it is the first to achieve the asymptotically optimal depth of $O(\log (n/\epsilon))$ with a sublinear number of ancilla qubits. The reduction uses long-range gates but no measurements.
Question 156multiple-choice
Quantum error-correcting codes that are covariant with respect to unitary group actions are important for fault-tolerant quantum computing but face significant resource and error reduction limitations. Theoretical results explore the trade-offs between error rates, subsystem dimensions, and group representation properties when designing such codes.
Which of the following best characterizes the lower bound result for SU(d_L)-covariant quantum codes regarding the scaling of physical subsystem dimension with logical system dimension for fixed n?
1) The physical subsystem dimension must grow exponentially with the logical system dimension.
2) The physical subsystem dimension can remain constant regardless of logical system size.
3) The error parameter can be made arbitrarily small without increasing subsystem dimension.
4) The scaling is always linear with respect to logical system dimension.
5) The lower bound only applies to codes without symmetry constraints.
6) The subsystem dimension decreases as logical system dimension increases.
7) The scaling is polynomial in the logical system dimension for all n.
✓ Correct Answer:
The correct answer is 1) The physical subsystem dimension must grow exponentially with the logical system dimension..
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entangled with alarge reference system. In such a situation, we require ϵ worst≲d−1 L. Bound (55)then asserts that, at constant n, the physical subsystem dimension must grow exponentially in the logical system dimension. From Theorem 6, we can further write ϵworst½SUðdLÞ-covariant code /C138 ≥1 2n1 max ilndiþO/C181 ndL/C19 ð56Þ (see proof in Supplemental Material, Sec. E. 2 [37]). The bound (56) is useful to determine the precision limit of a code that has a universal set of transversal gates. If weimagine that each physical subsystem is composed of m i¼log 2ðdiÞqubits lumped together, then the error parameter of the code scales at least inversely in the largestnumber of qubits m ithat are lumped together. If we consider, for instance, ten logical qubits ( dL¼210) that are encoded into nsystems consisting of ten qubits each, i.e.,di¼210, we obtain the rather prohibitive error param- eterϵworst≳0.14=n. In this example, we can improve this estimate to ϵworst≥0.5=nby using Eq. (54a) directly: If max idi¼dL, we must have ⌈½2nϵworst/C138−1⌉≤1, because any larger value would make the binomial coefficient too large to satisfy Eq. (54a) . B. Random constructions The bounds of Theorem 6 severely limit the error- correction capability of the unitary SUðdLÞ-covariant codes. We now show that it is possible to find good SUðdLÞ-covariant codes in regimes of large physical systems that are not excluded by Theorem 6. The constructions we present are randomized as well as asymptotic in the dimension of the physical subsystems.More precisely, we consider the encoding of one d L- dimensional Hilbert space HLin a physical space which is a tensor product of three Hilbert spaces HA¼HA1⊗ HA2⊗HA3. The encoding is done via an isometry VL→A, which is UðdLÞcovariant: For all U∈UðdLÞ, VU¼r1ðUÞ⊗r2ðUÞ⊗r3ðUÞV: ð57Þ Here, r1,r2, and r3are three irreps of UðdLÞ.O u r constructions are randomized in the following way: (i)Vis chosen randomly from all possible isometries satisfying the covariance condition (57);(ii) the irreps r1,r2, and r3are chosen randomly, or at least generically . In fact, we need only that the irreducible representation does not belong to a small subset of all possible irreducible representations. We use randomized constructions to prove the existence of UðdLÞ-covariant codes with a small error [measured by ϵe based on the (fixed-input) entanglement fidelity], as sum- marized in the following theorem. We refer to theSupplemental Material, Sec. F [37] for a complete proof. Theorem 7. Ford L≥4and every ϵ>0, there exists a UðdLÞ-covariant code with error ϵe≤ϵand physical dimensions di,i∈f1;2;3g, such that max ilndi≤dLðdL−1Þln/C181 ϵe/C19 þC2; ð58Þ for some C2which is only a function of dL. It is not clear how to compare the performance of our code given by Eq. (58) to our bounds of Theorem 6, because our nonconstructive proof does not specify the behavior of C2as a function of dL, which is given by details of the representation theory of UðdLÞ. It remains open whether the lower bound can be strengthened or theconstructions can be improved. Our proof technique does not immediately work for Uð2Þ-covariant codes, as it is harder to
Question 157multiple-choice
Quantum error correction is fundamental for the reliable operation of quantum computers, which are vulnerable to noise and decoherence. Stabilizer codes represent a versatile class of quantum error-correcting codes enabling robust protection against certain types of quantum errors.
Which quantum code is the smallest stabilizer code capable of correcting arbitrary single-qubit errors?
1) Nine-qubit Shor code
2) Five-qubit code
3) Seven-qubit Steane code
4) Ten-qubit Bacon-Shor code
5) Four-qubit code
6) Surface code
7) Color code
✓ Correct Answer:
The correct answer is 2) Five-qubit code.
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it was soon recognized that error-correction techniques could play an important role in a functioning quantum computer. Semi nal papers by Shor [64] and Steane [68,69] suggested a methodology, and a theory of quantum error-correcting codes was soon in place. In Chapter 4 we trace the development of this class of algorithms, beginning with a brief discussion of the physical problem of decoherence and discussing the early models of Shor and Steane. A "truly quantum" five- "qubit" code is discussed in detail, since it provides the paradigm for the "stabilizer" codes which are described subsequently. The relation of classical codes to stabilizer codes is recorded, and we conclude with a discussion of error correcting codes from the abstract Hilbert space perspective a la Knill and Laflamme [47]. There is much more to the subject of quantum computation and re lated fields than we discuss here. For example, in 1984 Charles Bennett and Gilles Brassard [10] introduced a key distribution system based on quantum mechanical principles. (For a novel extension of the ideas in [10], one which has a practical experimental realization, see Ekert [81].) Without going into detail, the goal is to enable sender and receiver (Alice and Bob) to securely construct a key sequence for encryption using the properties of photons. Thus, concurrently with quantum computers and quantum algorithms, the fields of quantum cryptography and quantum x Preface communication were developing. This quickly led to quantum information theory and to related complexity issues. The notation and results we present here provide a good foundation for further reading in these various subjects. A broad overview from the information-theoretic perspective is given in [70j, and readers interested in learning more about these various topics will find appropriate refer ences at the start of the bibliography section. Acknow ledgments An Introduction to Quantum Computing Algorithms reflects my own experience in learning the mathematics and theoretical physics required for the subject. I am most grateful to many colleagues who helped in that process, especially the core members of our Quantum Computation Seminar at the Baltimore County campus of the University of Maryland (UMBC) who comprised the (usually) patient audience for the initial presentation of parts of this book. Special thanks go to Rich Davis, Keith Miller, and Mort Rubin who read all or parts of the manuscript in its various incarnations. Readers who find additional errors are encouraged to notify me at pittenge@math.umbc.edu. I am also indebted to the many scholars who routinely made their research available on the Los Alamos web site quant-ph. The field has developed so quickly that an introduction of this nature would not have been possible otherwise. Ann Kostant at Birkhauser Boston was most supportive throughout the entire process, and indeed there would have been no book without her encouragement. Viet Ngo provided crucial support in preparing the various drawings, and I also received critical technical help in the prepara tion of the manuscript from Rouben Rostamian, Boris Alemi, and Esther M. Scheffel, colleagues at UMBC. That assistance
Question 158multiple-choice
Quantum approximate counting is a technique used to estimate the number of solutions (marked items) in a dataset using quantum algorithms. These algorithms often leverage quantum speedups and can be implemented using different subroutines such as Grover iterations and the Quantum Fourier Transform (QFT).
Which quantum algorithmic improvement allows for approximate counting and amplitude estimation using only Grover iterations, thereby eliminating the need for the Quantum Fourier Transform?
1) The use of quantum phase estimation based on QFT
2) A QFT-free algorithm that applies only Grover iterations for estimation
3) Classical probabilistic estimation with amplitude amplification
4) The integration of hidden subgroup algorithms
5) Quantum walk-based search algorithm
6) Variational quantum eigensolver for amplitude estimation
7) Quantum annealing with error correction
✓ Correct Answer:
The correct answer is 2) A QFT-free algorithm that applies only Grover iterations for estimation.
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Title: Quantum Approximate Counting, Simplified Year: 2019 Paper ID: d10c4b69506cbb8cbf0779f55d2284a9fe0a56e2 Source: semantic-scholar URL: https://www.semanticscholar.org/paper/d10c4b69506cbb8cbf0779f55d2284a9fe0a56e2 Abstract: In 1998, Brassard, Hoyer, Mosca, and Tapp (BHMT) gave a quantum algorithm for approximate counting. Given a list of $N$ items, $K$ of them marked, their algorithm estimates $K$ to within relative error $\varepsilon$ by making only $O\left( \frac{1}{\varepsilon}\sqrt{\frac{N}{K}}\right) $ queries. Although this speedup is of "Grover" type, the BHMT algorithm has the curious feature of relying on the Quantum Fourier Transform (QFT), more commonly associated with Shor's algorithm. Is this necessary? This paper presents a simplified algorithm, which we prove achieves the same query complexity using Grover iterations only. We also generalize this to a QFT-free algorithm for amplitude estimation. Related approaches to approximate counting were sketched previously by Grover, Abrams and Williams, Suzuki et al., and Wie (the latter two as we were writing this paper), but in all cases without rigorous analysis.
Question 159multiple-choice
In image processing, quantitative metrics are essential for evaluating the fidelity of processed images compared to their original versions. Two widely used metrics are Peak Signal-to-Noise Ratio (PSNR) and Structural Similarity Index (SSIM), which assess image quality from different perspectives.
Which of the following statements best describes a fundamental difference between PSNR and SSIM when used to evaluate image quality?
1) PSNR measures color accuracy, while SSIM measures noise level.
2) PSNR is calculated using histograms, whereas SSIM uses edge detection.
3) PSNR directly models human perception, while SSIM is a purely mathematical metric.
4) PSNR is suitable only for grayscale images, but SSIM works for color images.
5) PSNR reports values in percentages, whereas SSIM reports values in decibels.
6) PSNR evaluates pixel-wise intensity differences, while SSIM incorporates luminance, contrast, and structural information.
7) PSNR quantifies average pixel-wise error, but SSIM assesses perceptual similarity by considering luminance, contrast, and structure.
✓ Correct Answer:
The correct answer is 7) PSNR quantifies average pixel-wise error, but SSIM assesses perceptual similarity by considering luminance, contrast, and structure..